CEE 260 / MIE 273 Probability & Statistics Name: Final Exam, version D — 100 points (120 minutes) PLEASE READ QUESTIONS CAREFULLY and SHOW YOUR WORK! CALCULATORS PERMITTED – ABSOLUTELY NO REFERENCES! 1. Suppose the waiting time (in minutes) for your 911 SC Targa to reach operating temperature in the morning is uniformly distributed on [0,10], while the waiting time in the evening is uniformly distributed on [0,5] independent of morning waiting time. a. (5%) If you drive your Targa each morning and evening for a week (5 morning and 5 evening rides), what is the variance of your total waiting time? b. (5%) What is the expected value of the difference between morning and evening waiting time on a given day? 2. (10%) Find the maximum likelihood estimator (MLE) of ϴ when Xi ~ Exponential(ϴ) and you have observed X1, X2, X3, …, Xn. 2 3. The waiting time for delivery of a new Porsche 911 Carrera at the local dealership is distributed exponentially with a population mean of 3.55 months and population standard deviation of 1.1 months. Recently, in an effort to reduce the waiting time, the dealership has experimented with an online ordering system. A sample of 100 customers during a recent sales promotion generated a mean waiting time of 3.18 months using the new system. Assume that the population standard deviation of the waiting time has not changed from 1.1 months. (hint: the source distribution is irrelevant, but its parameters are relevant) a. (15%) What is the probability that the average wait time is between 3.2 and 6.4 months? (hint: draw a sketch for full credit) b. (10%) At the 0.05 level of significance, using the critical values approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? c. (10%) At the 0.01 level of significance, using the p-value approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? 3 4. Porsche AG is a leading manufacturer of performance automobiles. The 911 Carrera model, Porsche’s premier sports car, reaches a top track speed of 180 miles per hour. Engineers claim the new advanced technology 911 GT2 automatically adjusts its top speed depending on the weather conditions. Suppose that in an effort to test this claim, Porsche selects a few 911 GT2 models to test drive on the company track in Stuttgart, Germany. The average top speed for the sample of 25 test drives is 182.36 mph, with a standard deviation of 7.24 mph. a. (5%) Without using complete sentences, what might be some problems with the sampling conducted above? Identify and explain at least 2. b. (15%) Using the critical values approach to hypothesis testing and a 0.10 level of significance, is there evidence that the mean top track speed is different for the 911 GT2? (hint: state the null and alternative hypotheses, draw a sketch, and show your work for full credit) c. (10%) Set up a 95% confidence interval estimate of the population mean top speed of the 911 GT2. d. (5%) Compare the results of (b) and (c). What conclusions do you reach about the top speed of the new 911 GT2? 4 5. (10%) Porsche USA believes that sales of the venerable 911 Carrera are a function of annual income (in thousands of dollars) and a risk tolerance index of the potential buyer. Determine the regression equation and provide a succinct analysis of Porsche’s conjecture using the following Excel results. SUMMARY OUTPUT Regression Stat istics Multiple R 0.805073 R Square 0.648142 Adjusted R Square 0.606747 Standard Error 7.76312 Observations 20 ANOVA df SS MS F Significance F Regression 2 1887.227445 943.6137225 15.65747206 0.000139355 Residual 17 1024.522555 60.26603265 Total 19 2911.75 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 23.50557 6.845545641 3.433702952 0.003167982 9.062731576 37.94840898 Income 0.613408 0.125421229 4.890786567 0.000137795 0.348792801 0.878024121 Risk Index -0.00126 0.004519817 -0.278357691 0.784095184 -0.010794106 0.008277854 BONUS (5 points) What is the probability that 2 or more students in our class of 22 have the same birthday?

CEE 260 / MIE 273 Probability & Statistics Name: Final Exam, version D — 100 points (120 minutes) PLEASE READ QUESTIONS CAREFULLY and SHOW YOUR WORK! CALCULATORS PERMITTED – ABSOLUTELY NO REFERENCES! 1. Suppose the waiting time (in minutes) for your 911 SC Targa to reach operating temperature in the morning is uniformly distributed on [0,10], while the waiting time in the evening is uniformly distributed on [0,5] independent of morning waiting time. a. (5%) If you drive your Targa each morning and evening for a week (5 morning and 5 evening rides), what is the variance of your total waiting time? b. (5%) What is the expected value of the difference between morning and evening waiting time on a given day? 2. (10%) Find the maximum likelihood estimator (MLE) of ϴ when Xi ~ Exponential(ϴ) and you have observed X1, X2, X3, …, Xn. 2 3. The waiting time for delivery of a new Porsche 911 Carrera at the local dealership is distributed exponentially with a population mean of 3.55 months and population standard deviation of 1.1 months. Recently, in an effort to reduce the waiting time, the dealership has experimented with an online ordering system. A sample of 100 customers during a recent sales promotion generated a mean waiting time of 3.18 months using the new system. Assume that the population standard deviation of the waiting time has not changed from 1.1 months. (hint: the source distribution is irrelevant, but its parameters are relevant) a. (15%) What is the probability that the average wait time is between 3.2 and 6.4 months? (hint: draw a sketch for full credit) b. (10%) At the 0.05 level of significance, using the critical values approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? c. (10%) At the 0.01 level of significance, using the p-value approach to hypothesis testing, is there evidence that the population mean waiting time to accept delivery is less than 3.55 months? 3 4. Porsche AG is a leading manufacturer of performance automobiles. The 911 Carrera model, Porsche’s premier sports car, reaches a top track speed of 180 miles per hour. Engineers claim the new advanced technology 911 GT2 automatically adjusts its top speed depending on the weather conditions. Suppose that in an effort to test this claim, Porsche selects a few 911 GT2 models to test drive on the company track in Stuttgart, Germany. The average top speed for the sample of 25 test drives is 182.36 mph, with a standard deviation of 7.24 mph. a. (5%) Without using complete sentences, what might be some problems with the sampling conducted above? Identify and explain at least 2. b. (15%) Using the critical values approach to hypothesis testing and a 0.10 level of significance, is there evidence that the mean top track speed is different for the 911 GT2? (hint: state the null and alternative hypotheses, draw a sketch, and show your work for full credit) c. (10%) Set up a 95% confidence interval estimate of the population mean top speed of the 911 GT2. d. (5%) Compare the results of (b) and (c). What conclusions do you reach about the top speed of the new 911 GT2? 4 5. (10%) Porsche USA believes that sales of the venerable 911 Carrera are a function of annual income (in thousands of dollars) and a risk tolerance index of the potential buyer. Determine the regression equation and provide a succinct analysis of Porsche’s conjecture using the following Excel results. SUMMARY OUTPUT Regression Stat istics Multiple R 0.805073 R Square 0.648142 Adjusted R Square 0.606747 Standard Error 7.76312 Observations 20 ANOVA df SS MS F Significance F Regression 2 1887.227445 943.6137225 15.65747206 0.000139355 Residual 17 1024.522555 60.26603265 Total 19 2911.75 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 23.50557 6.845545641 3.433702952 0.003167982 9.062731576 37.94840898 Income 0.613408 0.125421229 4.890786567 0.000137795 0.348792801 0.878024121 Risk Index -0.00126 0.004519817 -0.278357691 0.784095184 -0.010794106 0.008277854 BONUS (5 points) What is the probability that 2 or more students in our class of 22 have the same birthday?

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The following equation can be used to compute values of y as a function of x: y = b  e?ax  sin(b  x)  (0:012  x4 ? 0:15  x3 + 0:075  x2 + 2:5  x) where a and b are parameters. Write the equation for implementation with MATLAB, where a = 2, b = 5, and x is a vector holding values from 0 to =24 in increments of x = =40. Employ the minimum number of periods (i.e., dot notation) so that your formulation yields a vector for y. In addition, compute the vector z = y2 where each element holds the square of each element of y. Combine x, y, and z into a matrix w, where each column holds one of the variables, and display w using the short g format. In addition, generate a labeled plot of y and z versus x. Include a legend on the plot (use help to understand how to do this). For y, use a 1:5-point, dashdotted red line with 14-point, red-edged white-faced pentagram-shaped markers. For z, use a standard-sized (i.e., default) solid blue line with standard-sized, blue-edged, green-faced square markers.

The following equation can be used to compute values of y as a function of x: y = b  e?ax  sin(b  x)  (0:012  x4 ? 0:15  x3 + 0:075  x2 + 2:5  x) where a and b are parameters. Write the equation for implementation with MATLAB, where a = 2, b = 5, and x is a vector holding values from 0 to =24 in increments of x = =40. Employ the minimum number of periods (i.e., dot notation) so that your formulation yields a vector for y. In addition, compute the vector z = y2 where each element holds the square of each element of y. Combine x, y, and z into a matrix w, where each column holds one of the variables, and display w using the short g format. In addition, generate a labeled plot of y and z versus x. Include a legend on the plot (use help to understand how to do this). For y, use a 1:5-point, dashdotted red line with 14-point, red-edged white-faced pentagram-shaped markers. For z, use a standard-sized (i.e., default) solid blue line with standard-sized, blue-edged, green-faced square markers.

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Homework 1 Q.1 what is product of these matrices? [■(3&2&6@-2&1&0@4&1&0)] ■(2@3@4) 6 Q.2 what is largest number of pivots a 7×5 matrix can have? Q.3 find standard matrix of linear transformation T: R^2→R^2 which rotates a point about the origion through an angle of π/4 radians. Q.4 True or False If matrices A and B are equivalent, they have same reduced echelon form In general A+B≠B+A I matrix A is symmetric then so is A+I_n A matrix A must be a square matrix to be invertible. If det (A) ≠ 0 then column of A are linearly independent. If n × n matrix is equivalent to I_n then A^(-1) is also equivalent toI_n. If m × n matrix has pivot position in every row then the equation Ax = b has a unique solution for each b in R^m If AB = I, then I is invertible.

Homework 1 Q.1 what is product of these matrices? [■(3&2&6@-2&1&0@4&1&0)] ■(2@3@4) 6 Q.2 what is largest number of pivots a 7×5 matrix can have? Q.3 find standard matrix of linear transformation T: R^2→R^2 which rotates a point about the origion through an angle of π/4 radians. Q.4 True or False If matrices A and B are equivalent, they have same reduced echelon form In general A+B≠B+A I matrix A is symmetric then so is A+I_n A matrix A must be a square matrix to be invertible. If det (A) ≠ 0 then column of A are linearly independent. If n × n matrix is equivalent to I_n then A^(-1) is also equivalent toI_n. If m × n matrix has pivot position in every row then the equation Ax = b has a unique solution for each b in R^m If AB = I, then I is invertible.

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1 Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3.1 Laboratory Objective The objective of this laboratory is to understand the basic properties of sinusoids and sinusoid measurements. 3.2 Educational Objectives After performing this experiment, students should be able to: 1. Understand the properties of sinusoids. 2. Understand sinusoidal manipulation 3. Use a function generator 4. Obtain measurements using an oscilloscope 3.3 Background Sinusoids are sine or cosine waveforms that can describe many engineering phenomena. Any oscillatory motion can be described using sinusoids. Many types of electrical signals such as square, triangle, and sawtooth waves are modeled using sinusoids. Their manipulation incurs the understanding of certain quantities that describe sinusoidal behavior. These quantities are described below. 3.3.1 Sinusoid Characteristics Amplitude The amplitude A of a sine wave describes the height of the hills and valleys of a sinusoid. It carries the physical units of what the sinusoid is describing (volts, amps, meters, etc.). Frequency There are two types of frequencies that can describe a sinusoid. The normal frequency f is how many times the sinusoid repeats per unit time. It has units of cycles per second (s-1) or Hertz (Hz). The angular frequency ω is how many radians pass per second. Consequently, ω has units of radians per second. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 2 Period The period T is how long a sinusoid takes to repeat one complete cycle. The period is measured in seconds. Phase The phase φ of a sinusoid causes a horizontal shift along the t-axis. The phase has units of radians. TimeShift The time shift ts of a sinusoid is a horizontal shift along the t-axis and is a time measurement of the phase. The time shift has units of seconds. NOTE: A sine wave and a cosine wave only differ by a phase shift of 90° or ?2 radians. In reality, they are the same waveform but with a different φ value. 3.3.2 Sinusoidal Relationships Figure 3.1: Sinusoid The general equation of a sinusoid is given below and refers to Figure 3.1. ?(?) = ????(?? +?) (3.1) The angular frequency is related to the normal frequency by Equation 3.2. ?= 2?? (3.2) The angular frequency is also related to the period by Equation 3.3. ?=2?? (3.3) By inspection, the normal frequency is related to the period by Equation 3.4. ? =1? (3.4) ?? Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3 The time shift is related to the phase (radians) and the frequency by Equation 3.5. ??= ∅2?? (3.5) 3.3.3 Equipment 3.3.3.1 Inductors Inductors are electrical components that resist a change in the flow of current passing through them. They are essentially coils of wire. Inductors are electromagnets too. They are represented in schematics using the following symbol and physically using the following equipment (with or without exposed wire): Figure 3.2: Symbol and Physical Example for Inductors 3.3.3.2 Capacitors Capacitors are electrical components that store energy. This enables engineers to store electrical energy from an input source such as a battery. Some capacitors are polarized and therefore have a negative and positive plate. One plate is straight, representing the positive terminal on the device, and the other is curved, representing the negative one. Polarized capacitors are represented in schematics using the following symbol and physically using the following equipment: Figure 3.3: Symbol and Physical Example for Capacitors 3.3.3.3 Function Generator A function generator is used to create different types of electrical waveforms over a wide range of frequencies. It generates standard sine, square, and triangle waveforms and uses the analog output channel. 3.3.3.5 Oscilloscope An oscilloscope is a type of electronic test instrument that allows observation of constantly varying voltages, usually as a two-dimensional plot of one or more signals as a function of time. It displays voltage data over time for the analysis of one or two voltage measurements taken from the analog input channels of the Oscilloscope. The observed waveform can be analyzed for amplitude, frequency, time interval and more. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 4 3.4 Procedure Follow the steps outlined below after the instructor has explained how to use the laboratory equipment 3.4.1 Sinusoidal Measurements 1. Connect the output channel of the Function Generator to the channel one of the Oscilloscope. 2. Complete Table 3.1 using the given values for voltage and frequency. Table 3.1: Sinusoid Measurements Function Generator Oscilloscope (Measured) Calculated Voltage Amplitude, A (V ) Frequency (Hz) 2*A (Vp−p ) f (Hz) T (sec) ω (rad/sec) T (sec) 2.5 1000 3 5000 3.4.2 Circuit Measurements 1. Connect the circuit in figure 3.4 below with the given resistor and capacitor NOTE: Vs from the circuit comes from the Function Generator using a BNC connector. Figure 3.4: RC Circuit Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 5 2. Using the alligator to BNC cables, connect channel one of the Oscilloscope across the capacitor and complete Table 3.2 Table 3.2: Capacitor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) Vc (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 3. Disconnect channel one and connect channel two of the oscilloscope across the resistor and complete table 3.3. Table 3.3: Resistor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) VR (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 4. Leaving channel two connected across the resistor, clip the positive lead to the positive side of the capacitor and complete table 3.4 Table 3.4: Phase Difference Function Generator Oscilloscope (Measured) Calculated Vs (volts) Frequency (Hz) Divisions Time/Div (sec) ts (sec) ɸ (rad) ɸ (degrees) 2.5 100 5. Using the data from Tables 3.2, 3.3, and 3.4, plot the capacitor sinusoidal equation and the resistor sinusoidal equation on the same graph using MATLAB. HINT: Plot over one period. 6. Kirchoff’s Voltage Law states that ??(?)=??(?)+??(?). Calculate Vs by hand using the following equation and Tables 3.2 and 3.3 ??(?)=√??2+??2???(??−???−1(????)) Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 6 3.5 New MATLAB Commands hold on  This command allows multiple graphs to be placed on the same XY axis and is placed after the first plot statement. legend (’string 1’, ’string2’, ‘string3’)  This command adds a legend to the plot. Strings must be placed in the order as the plots were generated. plot (x, y, ‘line specifiers’)  This command plots the data and uses line specifiers to differentiate between different plots on the same XY axis. In this lab, only use different line styles from the table below. Table 3.5: Line specifiers for the plot() command sqrt(X)  This command produces the square root of the elements of X. NOTE: The “help” command in MATLAB can be used to find a description and example for functions such as input.  For example, type “help input” in the command window to learn more about the input function. NOTE: Refer to section the “MATLAB Commands” sections from prior labs for previously discussed material that you may also need in order to complete this assignment. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 7 3.6 Lab Report Requirements 1. Complete Tables 3.1, 3.2, 3.3, 3.4 (5 points each) 2. Show hand calculations for all four tables. Insert after this page (5 points each) 3. Draw the two sinusoids by hand from table 3.1. Label amplitude, period, and phase. Insert after this page. (5 points) 4. Insert MATLAB plot of Vc and VR as obtained from data in Tables 3.2 and 3.3 after this page. (5 points each) 5. Show hand calculations for Vs(t). Insert after this page. (5 points) 6. Using the data from the Tables, write: (10 points) a) Vc(t) = b) VR(t) = 7. Also, ???(?)=2.5???(628?). Write your Vs below and give reasons why they are different. (10 points) a) Vs(t) = b) Reasons: 8. Write an executive summary for this lab describing what you have done, and learned. (20 points)

1 Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3.1 Laboratory Objective The objective of this laboratory is to understand the basic properties of sinusoids and sinusoid measurements. 3.2 Educational Objectives After performing this experiment, students should be able to: 1. Understand the properties of sinusoids. 2. Understand sinusoidal manipulation 3. Use a function generator 4. Obtain measurements using an oscilloscope 3.3 Background Sinusoids are sine or cosine waveforms that can describe many engineering phenomena. Any oscillatory motion can be described using sinusoids. Many types of electrical signals such as square, triangle, and sawtooth waves are modeled using sinusoids. Their manipulation incurs the understanding of certain quantities that describe sinusoidal behavior. These quantities are described below. 3.3.1 Sinusoid Characteristics Amplitude The amplitude A of a sine wave describes the height of the hills and valleys of a sinusoid. It carries the physical units of what the sinusoid is describing (volts, amps, meters, etc.). Frequency There are two types of frequencies that can describe a sinusoid. The normal frequency f is how many times the sinusoid repeats per unit time. It has units of cycles per second (s-1) or Hertz (Hz). The angular frequency ω is how many radians pass per second. Consequently, ω has units of radians per second. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 2 Period The period T is how long a sinusoid takes to repeat one complete cycle. The period is measured in seconds. Phase The phase φ of a sinusoid causes a horizontal shift along the t-axis. The phase has units of radians. TimeShift The time shift ts of a sinusoid is a horizontal shift along the t-axis and is a time measurement of the phase. The time shift has units of seconds. NOTE: A sine wave and a cosine wave only differ by a phase shift of 90° or ?2 radians. In reality, they are the same waveform but with a different φ value. 3.3.2 Sinusoidal Relationships Figure 3.1: Sinusoid The general equation of a sinusoid is given below and refers to Figure 3.1. ?(?) = ????(?? +?) (3.1) The angular frequency is related to the normal frequency by Equation 3.2. ?= 2?? (3.2) The angular frequency is also related to the period by Equation 3.3. ?=2?? (3.3) By inspection, the normal frequency is related to the period by Equation 3.4. ? =1? (3.4) ?? Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 3 The time shift is related to the phase (radians) and the frequency by Equation 3.5. ??= ∅2?? (3.5) 3.3.3 Equipment 3.3.3.1 Inductors Inductors are electrical components that resist a change in the flow of current passing through them. They are essentially coils of wire. Inductors are electromagnets too. They are represented in schematics using the following symbol and physically using the following equipment (with or without exposed wire): Figure 3.2: Symbol and Physical Example for Inductors 3.3.3.2 Capacitors Capacitors are electrical components that store energy. This enables engineers to store electrical energy from an input source such as a battery. Some capacitors are polarized and therefore have a negative and positive plate. One plate is straight, representing the positive terminal on the device, and the other is curved, representing the negative one. Polarized capacitors are represented in schematics using the following symbol and physically using the following equipment: Figure 3.3: Symbol and Physical Example for Capacitors 3.3.3.3 Function Generator A function generator is used to create different types of electrical waveforms over a wide range of frequencies. It generates standard sine, square, and triangle waveforms and uses the analog output channel. 3.3.3.5 Oscilloscope An oscilloscope is a type of electronic test instrument that allows observation of constantly varying voltages, usually as a two-dimensional plot of one or more signals as a function of time. It displays voltage data over time for the analysis of one or two voltage measurements taken from the analog input channels of the Oscilloscope. The observed waveform can be analyzed for amplitude, frequency, time interval and more. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 4 3.4 Procedure Follow the steps outlined below after the instructor has explained how to use the laboratory equipment 3.4.1 Sinusoidal Measurements 1. Connect the output channel of the Function Generator to the channel one of the Oscilloscope. 2. Complete Table 3.1 using the given values for voltage and frequency. Table 3.1: Sinusoid Measurements Function Generator Oscilloscope (Measured) Calculated Voltage Amplitude, A (V ) Frequency (Hz) 2*A (Vp−p ) f (Hz) T (sec) ω (rad/sec) T (sec) 2.5 1000 3 5000 3.4.2 Circuit Measurements 1. Connect the circuit in figure 3.4 below with the given resistor and capacitor NOTE: Vs from the circuit comes from the Function Generator using a BNC connector. Figure 3.4: RC Circuit Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 5 2. Using the alligator to BNC cables, connect channel one of the Oscilloscope across the capacitor and complete Table 3.2 Table 3.2: Capacitor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) Vc (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 3. Disconnect channel one and connect channel two of the oscilloscope across the resistor and complete table 3.3. Table 3.3: Resistor Sinusoid Function Generator Oscilloscope (Measured) Calculated Vs (Volts) Frequency (Hz) VR (volts) f (Hz) T (sec) ω (rad/sec) 2.5 100 4. Leaving channel two connected across the resistor, clip the positive lead to the positive side of the capacitor and complete table 3.4 Table 3.4: Phase Difference Function Generator Oscilloscope (Measured) Calculated Vs (volts) Frequency (Hz) Divisions Time/Div (sec) ts (sec) ɸ (rad) ɸ (degrees) 2.5 100 5. Using the data from Tables 3.2, 3.3, and 3.4, plot the capacitor sinusoidal equation and the resistor sinusoidal equation on the same graph using MATLAB. HINT: Plot over one period. 6. Kirchoff’s Voltage Law states that ??(?)=??(?)+??(?). Calculate Vs by hand using the following equation and Tables 3.2 and 3.3 ??(?)=√??2+??2???(??−???−1(????)) Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 6 3.5 New MATLAB Commands hold on  This command allows multiple graphs to be placed on the same XY axis and is placed after the first plot statement. legend (’string 1’, ’string2’, ‘string3’)  This command adds a legend to the plot. Strings must be placed in the order as the plots were generated. plot (x, y, ‘line specifiers’)  This command plots the data and uses line specifiers to differentiate between different plots on the same XY axis. In this lab, only use different line styles from the table below. Table 3.5: Line specifiers for the plot() command sqrt(X)  This command produces the square root of the elements of X. NOTE: The “help” command in MATLAB can be used to find a description and example for functions such as input.  For example, type “help input” in the command window to learn more about the input function. NOTE: Refer to section the “MATLAB Commands” sections from prior labs for previously discussed material that you may also need in order to complete this assignment. Laboratory 3 – Sinusoids in Engineering: Measurement and Analysis of Harmonic Signals 7 3.6 Lab Report Requirements 1. Complete Tables 3.1, 3.2, 3.3, 3.4 (5 points each) 2. Show hand calculations for all four tables. Insert after this page (5 points each) 3. Draw the two sinusoids by hand from table 3.1. Label amplitude, period, and phase. Insert after this page. (5 points) 4. Insert MATLAB plot of Vc and VR as obtained from data in Tables 3.2 and 3.3 after this page. (5 points each) 5. Show hand calculations for Vs(t). Insert after this page. (5 points) 6. Using the data from the Tables, write: (10 points) a) Vc(t) = b) VR(t) = 7. Also, ???(?)=2.5???(628?). Write your Vs below and give reasons why they are different. (10 points) a) Vs(t) = b) Reasons: 8. Write an executive summary for this lab describing what you have done, and learned. (20 points)

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For Day 19 Homework Cover Sheet Name:_________________________________________________ Read Pages from 294-315, or watch the videos listed below Introduction to Division http://www.youtube.com/watch?v=7gZ4yW1nr9Y (13 min) Introduction to Division of Rational Numbers http://www.youtube.com/watch?v=9LTICGxqwKE (10 min) Division of Decimal Numbers and Rational Expressions http://www.youtube.com/watch?v=BGReDOGObbk (7 min) Division Algorithm for Decimal Numbers and Polynomials http://www.youtube.com/watch?v=XXr0ixy8PfA (8 min) Division Algorithm for Decimal Polynomials http://www.youtube.com/watch?v=PQrlt8PhFAE (11 min) Attempt problems from workbook pages 91-97 Summary of the lectures you watched should include answers to the following questions. When doing division of rational expressions or numbers, what allows us to multiply by the reciprocal of the divisor instead? For example, 3/4÷3/5=3/4×5/3 or (x^2-1)/x÷(x+1)/(x-2)=(x^2-1)/x×(x-2)/(x-1) List any parts of the video lecture (if there are any) that were unclear or you had trouble understanding. Please be specific and do not just say “All of it”. Questions you had difficulty with or felt stuck on- List the number for the ALEKS topics you were stuck on from the list at the end of the video logs-   ALEKS Topics Mastered Division facts Division involving a whole number and a fraction Division involving quotients with intermediate zeros Division of a decimal by a power of ten Division with carry Division with trailing zeros: Problem type 1 Division with trailing zeros: Problem type 2 Division without carry Fraction division Integer multiplication and division Multiplying or dividing numbers written in scientific notation Quotient and remainder: Problem type 1 Quotient and remainder: Problem type 2 Quotient and remainder: Problem type 3 Rationalizing the denominator of a radical expression Simplifying a product and quotient involving square roots of negative numbers The reciprocal of a number Writing a ratio as a percentage without a calculator

For Day 19 Homework Cover Sheet Name:_________________________________________________ Read Pages from 294-315, or watch the videos listed below Introduction to Division http://www.youtube.com/watch?v=7gZ4yW1nr9Y (13 min) Introduction to Division of Rational Numbers http://www.youtube.com/watch?v=9LTICGxqwKE (10 min) Division of Decimal Numbers and Rational Expressions http://www.youtube.com/watch?v=BGReDOGObbk (7 min) Division Algorithm for Decimal Numbers and Polynomials http://www.youtube.com/watch?v=XXr0ixy8PfA (8 min) Division Algorithm for Decimal Polynomials http://www.youtube.com/watch?v=PQrlt8PhFAE (11 min) Attempt problems from workbook pages 91-97 Summary of the lectures you watched should include answers to the following questions. When doing division of rational expressions or numbers, what allows us to multiply by the reciprocal of the divisor instead? For example, 3/4÷3/5=3/4×5/3 or (x^2-1)/x÷(x+1)/(x-2)=(x^2-1)/x×(x-2)/(x-1) List any parts of the video lecture (if there are any) that were unclear or you had trouble understanding. Please be specific and do not just say “All of it”. Questions you had difficulty with or felt stuck on- List the number for the ALEKS topics you were stuck on from the list at the end of the video logs-   ALEKS Topics Mastered Division facts Division involving a whole number and a fraction Division involving quotients with intermediate zeros Division of a decimal by a power of ten Division with carry Division with trailing zeros: Problem type 1 Division with trailing zeros: Problem type 2 Division without carry Fraction division Integer multiplication and division Multiplying or dividing numbers written in scientific notation Quotient and remainder: Problem type 1 Quotient and remainder: Problem type 2 Quotient and remainder: Problem type 3 Rationalizing the denominator of a radical expression Simplifying a product and quotient involving square roots of negative numbers The reciprocal of a number Writing a ratio as a percentage without a calculator

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Comp755 HW 2 – Fall 2015 1. Problem 4.2 (10pts) 2. Problem 4.4 (15pts) 3. Problem 5.25 (10pts) 4. Problem 10.2 (15pts) . Hint: Create three Gantt charts where each square represents 10 time units. a. The first chart should use earliest deadline using the processes that are currently available. Since there is no preemption, some processes will not be scheduled if there start deadline is missed. b. The second chart should schedule strictly by the earliest deadline. The processor will be idle if the process with the earliest deadline has not arrived. c. The third chart should just use FCFS. 5. (Synchronize threads) Write a program that launches 1,000 threads. Each thread adds 1 to a variable sum that initially is 0. You need to pass sum by reference to each thread. In order to pass it by reference, define an Integer wrapper object to hold sum. Run the program with and without synchronization to see its effect (Create a command line argument where passing a 0 means run unsynchronized and passing a 1 means to run synchronized). Submit your entire Netbeans project (50pts).

Comp755 HW 2 – Fall 2015 1. Problem 4.2 (10pts) 2. Problem 4.4 (15pts) 3. Problem 5.25 (10pts) 4. Problem 10.2 (15pts) . Hint: Create three Gantt charts where each square represents 10 time units. a. The first chart should use earliest deadline using the processes that are currently available. Since there is no preemption, some processes will not be scheduled if there start deadline is missed. b. The second chart should schedule strictly by the earliest deadline. The processor will be idle if the process with the earliest deadline has not arrived. c. The third chart should just use FCFS. 5. (Synchronize threads) Write a program that launches 1,000 threads. Each thread adds 1 to a variable sum that initially is 0. You need to pass sum by reference to each thread. In order to pass it by reference, define an Integer wrapper object to hold sum. Run the program with and without synchronization to see its effect (Create a command line argument where passing a 0 means run unsynchronized and passing a 1 means to run synchronized). Submit your entire Netbeans project (50pts).

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A square loop of wire with a small resistance is moved with constant speed from a field free region into a region of uniform B field (B is constant in time) and then back into a field free region to the left. The self inductance of loop is negligible. QUESTION: A) When leaving the field the coil experiences a magnetic force to the left. B) While the loop is entirely in the field, the emf in the loop is zero. C) When entering the field the coil experiences a magnetic force to the right. D) Upon leaving the field, a clockwise current flows in the loop. E) Upon entering the field, a clockwise current flows in the loop.

A square loop of wire with a small resistance is moved with constant speed from a field free region into a region of uniform B field (B is constant in time) and then back into a field free region to the left. The self inductance of loop is negligible. QUESTION: A) When leaving the field the coil experiences a magnetic force to the left. B) While the loop is entirely in the field, the emf in the loop is zero. C) When entering the field the coil experiences a magnetic force to the right. D) Upon leaving the field, a clockwise current flows in the loop. E) Upon entering the field, a clockwise current flows in the loop.

A six- turn square loop of sides 0.2 m lies in a plane perpendicular to a uniform magnetic field of magnitude 0.5 T. If the wire reshaped into a three- turn square of sides 0.4 m in 0.1 sec, what is the magnitude of the emf induced in wire during this time ? 1) 0.68 volts 2) 1.2 volts 3) 2.4 volts 4) 3.6 volts 5) 4.8 volts.

A six- turn square loop of sides 0.2 m lies in a plane perpendicular to a uniform magnetic field of magnitude 0.5 T. If the wire reshaped into a three- turn square of sides 0.4 m in 0.1 sec, what is the magnitude of the emf induced in wire during this time ? 1) 0.68 volts 2) 1.2 volts 3) 2.4 volts 4) 3.6 volts 5) 4.8 volts.

 
Math 285 Quiz One Name: ________________ 1. The product of these two matrices 5 2 2 −1  4 −2 1 3  is (Please show your work for full credit.) 2. For what value of a is this determinant equal to 0? 0 5 2 0 1 −8 −4 2 (please justify your answer.) 3. What is the product of these matrices? 3 2 6 −2 1 0 4 1 0   2346  (Please justify your answer.) 4. What is the largest possible number of pivots a 7 × 5 matrix can have? (Please justify your answer.) Homework 1 5. Find the standard matrix of the linear transformation : →  which rotates a point about the origin through an angle of   radians (counterclockwise). True or False (Simply enter T or F, no need to justify the answer) If matrices  and  are row equivalent, they have the same reduced echelon form. In general,  +  ≠  +  (assume  and  are  x  matrices). If a matrix  is symmetric, then so is  + “#. A matrix  must be a square matrix to be invertible. If $%&’( ≠ 0, then columns of  are linearly independent. If an  x  matrix  is equivalent to “#, then )* is also equivalent to “#. If an + x  matrix  has a pivot position in every row, then the equation , = . has a unique solution for each . in /0. If  = “, then  is invertible.

Math 285 Quiz One Name: ________________ 1. The product of these two matrices 5 2 2 −1  4 −2 1 3  is (Please show your work for full credit.) 2. For what value of a is this determinant equal to 0? 0 5 2 0 1 −8 −4 2 (please justify your answer.) 3. What is the product of these matrices? 3 2 6 −2 1 0 4 1 0   2346  (Please justify your answer.) 4. What is the largest possible number of pivots a 7 × 5 matrix can have? (Please justify your answer.) Homework 1 5. Find the standard matrix of the linear transformation : →  which rotates a point about the origin through an angle of   radians (counterclockwise). True or False (Simply enter T or F, no need to justify the answer) If matrices  and  are row equivalent, they have the same reduced echelon form. In general,  +  ≠  +  (assume  and  are  x  matrices). If a matrix  is symmetric, then so is  + “#. A matrix  must be a square matrix to be invertible. If $%&’( ≠ 0, then columns of  are linearly independent. If an  x  matrix  is equivalent to “#, then )* is also equivalent to “#. If an + x  matrix  has a pivot position in every row, then the equation , = . has a unique solution for each . in /0. If  = “, then  is invertible.

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ENGR216: Mechanics and Vibrations Tutorial sheet 1 Michaelmas Term AY 2015/2016 Problems will be solved in class in week 5 PROBLEM 1 A rod of length L, cross-sectional area A1, and modulus of elasticity E1 has been placed inside a tube of the same length L, but of cross-sectional area A2 and modulus of elasticity E2. A force P is applied on a rigid plate attached to both tube and rod, as shown in the sketch below. Determine: a) the horizontal displacement of the rigid plate; b) the fixed support reactions acting on the rod and tube when E1=E2; c) the fixed support reactions acting on the rod and tube when E1=2E2; HINT: deformation of tube and rod is constrained to be the same. PROBLEM 2 A steel beam has a rectangular cross section of height lx=20 mm and width ly=30 mm, and length lz=1 m (lengths lx, ly and lz are measured respectively along x, y and z axes of a Cartesian system). The material of the beam has Young modulus E=200 GPa, Poisson ratio ν=0.29, and maximum allowable normal stress of 175 MPa. The beam is subject to a compressive centric axial load Pz of 80 KN applied at its ends (load acts along z axis). a) State whether the area of the cross section of the beam will increase or decrease under the effect of the applied centric axial load and explain why. b) Determine the variation of the section height lx in mm, indicating if such variation is a contraction or an elongation. c) Determine the maximum axial load (Pz)max applicable to the beam and the maximum shear stress in these conditions. d) In the loading condition (c), state whether the uniformly distributed normal load to be applied on the beam faces normal to the x axis leading to a zero variation of the section height lx is compressive or tensile and justify your answer. e) In the loading condition (c), determine the magnitude of the uniformly distributed normal load to be applied on the beam faces normal to the x axis resulting in zero variation of the section height lx. f) After application of the uniformly distributed normal load, determine the bulk modulus and the beam dilatation indicating its sign. PROBLEM 3 A beam has a constant circular cross section of radius 20 mm, and is subject to a tensile axial load of 4 KN. a) Determine the magnitude of the maximum stress in the cross section when the axial load is applied at the centre of the section. b) In the loading condition (a), state whether a neutral axis exists or not, and explain why. c) State whether the maximum stress in the cross section when the axial load is applied at 10 mm from the centre of the section is compressive or tensile and explain why. d) In the loading condition (c), determine the magnitude of the maximum compressive and tensile stresses in the cross section. e) In the loading condition (c), determine the distance of the neutral axis from the centre of the cross section. PROBLEM 4 Consider a simply supported beam subject to the distributed load sketched below. a) Determine the equations of shear force V(x) and bending-moment M(x); b) plot V(x) and M(x) along the beam axis; c) assuming the cross section is square and has length a , determine the position along the beam where the maximum normal stress occurs and the value of such maximum normal stress; d) determine the position along the beam where the maximum shear stress occurs and the value of such maximum shear stress.

ENGR216: Mechanics and Vibrations Tutorial sheet 1 Michaelmas Term AY 2015/2016 Problems will be solved in class in week 5 PROBLEM 1 A rod of length L, cross-sectional area A1, and modulus of elasticity E1 has been placed inside a tube of the same length L, but of cross-sectional area A2 and modulus of elasticity E2. A force P is applied on a rigid plate attached to both tube and rod, as shown in the sketch below. Determine: a) the horizontal displacement of the rigid plate; b) the fixed support reactions acting on the rod and tube when E1=E2; c) the fixed support reactions acting on the rod and tube when E1=2E2; HINT: deformation of tube and rod is constrained to be the same. PROBLEM 2 A steel beam has a rectangular cross section of height lx=20 mm and width ly=30 mm, and length lz=1 m (lengths lx, ly and lz are measured respectively along x, y and z axes of a Cartesian system). The material of the beam has Young modulus E=200 GPa, Poisson ratio ν=0.29, and maximum allowable normal stress of 175 MPa. The beam is subject to a compressive centric axial load Pz of 80 KN applied at its ends (load acts along z axis). a) State whether the area of the cross section of the beam will increase or decrease under the effect of the applied centric axial load and explain why. b) Determine the variation of the section height lx in mm, indicating if such variation is a contraction or an elongation. c) Determine the maximum axial load (Pz)max applicable to the beam and the maximum shear stress in these conditions. d) In the loading condition (c), state whether the uniformly distributed normal load to be applied on the beam faces normal to the x axis leading to a zero variation of the section height lx is compressive or tensile and justify your answer. e) In the loading condition (c), determine the magnitude of the uniformly distributed normal load to be applied on the beam faces normal to the x axis resulting in zero variation of the section height lx. f) After application of the uniformly distributed normal load, determine the bulk modulus and the beam dilatation indicating its sign. PROBLEM 3 A beam has a constant circular cross section of radius 20 mm, and is subject to a tensile axial load of 4 KN. a) Determine the magnitude of the maximum stress in the cross section when the axial load is applied at the centre of the section. b) In the loading condition (a), state whether a neutral axis exists or not, and explain why. c) State whether the maximum stress in the cross section when the axial load is applied at 10 mm from the centre of the section is compressive or tensile and explain why. d) In the loading condition (c), determine the magnitude of the maximum compressive and tensile stresses in the cross section. e) In the loading condition (c), determine the distance of the neutral axis from the centre of the cross section. PROBLEM 4 Consider a simply supported beam subject to the distributed load sketched below. a) Determine the equations of shear force V(x) and bending-moment M(x); b) plot V(x) and M(x) along the beam axis; c) assuming the cross section is square and has length a , determine the position along the beam where the maximum normal stress occurs and the value of such maximum normal stress; d) determine the position along the beam where the maximum shear stress occurs and the value of such maximum shear stress.

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