Explain the term “Stress Relaxation” and discuss its significance in the design of polymer components.

## Explain the term “Stress Relaxation” and discuss its significance in the design of polymer components.

Reduction in stress in a material subjected to continued constant … Read More...
Distribution of the Sample Mean and Linear Combinations – Examples Example 1 Let X1;X2; : : : ;X100 denote the actual net weights of 100 randomly selected 50-pound bags of fertilizer. a. If the expected weight of each bag is 50 pounds and the standard deviation is 1 pound, approximate P(49:75 • ¹X • 50:25) using the CLT. b. If the expected weight is 49.8 pounds rather than 50 pounds, so that on average bags are under…lled, approximate P(49:75 • ¹X • 50:25). Example 2 The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. a. What is the approximate probability that the sample mean breaking strength for a random sample of 40 rivets is between 9,900 psi and 10,200 psi? b. If the sample size had been 15 rivets rather than 40 rivets, could the probability requested in part a be approximated from the given information? Why or why not? Example 3 The lifetime of a certain type of battery is normally distributed with mean 8 hours and standard deviation 1 hour. There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages? Example 4 Suppose your waiting time for a bus in the morning is uniformly distributed on [0; 5], while waiting time in the evening is uniformly distributed on [0; 10]. Assume that evening waiting time is independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time. b. What is the variance of your total waiting time? expected value and variance of the di¤erence between morning and evening waiting time on a given day? d. What are the expected value and variance of the di¤erence between total morning waiting time and total evening waiting time for a particular week? 2 Example 5 Three di¤erent roads feed into a particular freeway entrance. Suppose that during a …xed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the following table: Road 1 Road 2 Road 3 Expected Value 800 1000 600 Standard Deviation 16 25 18 : a. What is the expected total number of cars entering the freeway at this point during the period? b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the number of cars on the di¤erent roads? c. With Xi denoting the number of cars entering from road i during the period, suppose that Cov(X1;X2) = 80, Cov(X1;X3) = 90, and Cov(X2;X3) = 100 (so that the three streams of tra¢c are not independent). Compute the expected total number of entering cars and the standard deviation of the total. Example 6 In an area having sandy soil, 50 small trees of a certain type were planted, and another 50 trees were planted in an area having clay soil. Let X be the number of trees planted in sandy soil that survive one year and Y be the number of trees planted in clay soil that survive one year. If the probability that a tree planted in sandy soil will survive one year is 0.7 and the probability of one-year survival in clay soil is 0.6, compute an approximation to P(¡5 • X ¡ Y • 5). For the purposes of this exercise, ignore the continuity correction.

## Distribution of the Sample Mean and Linear Combinations – Examples Example 1 Let X1;X2; : : : ;X100 denote the actual net weights of 100 randomly selected 50-pound bags of fertilizer. a. If the expected weight of each bag is 50 pounds and the standard deviation is 1 pound, approximate P(49:75 • ¹X • 50:25) using the CLT. b. If the expected weight is 49.8 pounds rather than 50 pounds, so that on average bags are under…lled, approximate P(49:75 • ¹X • 50:25). Example 2 The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. a. What is the approximate probability that the sample mean breaking strength for a random sample of 40 rivets is between 9,900 psi and 10,200 psi? b. If the sample size had been 15 rivets rather than 40 rivets, could the probability requested in part a be approximated from the given information? Why or why not? Example 3 The lifetime of a certain type of battery is normally distributed with mean 8 hours and standard deviation 1 hour. There are four batteries in a package. What lifetime value is such that the total lifetime of all batteries in a package exceeds that value for only 5% of all packages? Example 4 Suppose your waiting time for a bus in the morning is uniformly distributed on [0; 5], while waiting time in the evening is uniformly distributed on [0; 10]. Assume that evening waiting time is independent of morning waiting time. a. If you take the bus each morning and evening for a week, what is your total expected waiting time. b. What is the variance of your total waiting time? expected value and variance of the di¤erence between morning and evening waiting time on a given day? d. What are the expected value and variance of the di¤erence between total morning waiting time and total evening waiting time for a particular week? 2 Example 5 Three di¤erent roads feed into a particular freeway entrance. Suppose that during a …xed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the following table: Road 1 Road 2 Road 3 Expected Value 800 1000 600 Standard Deviation 16 25 18 : a. What is the expected total number of cars entering the freeway at this point during the period? b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the number of cars on the di¤erent roads? c. With Xi denoting the number of cars entering from road i during the period, suppose that Cov(X1;X2) = 80, Cov(X1;X3) = 90, and Cov(X2;X3) = 100 (so that the three streams of tra¢c are not independent). Compute the expected total number of entering cars and the standard deviation of the total. Example 6 In an area having sandy soil, 50 small trees of a certain type were planted, and another 50 trees were planted in an area having clay soil. Let X be the number of trees planted in sandy soil that survive one year and Y be the number of trees planted in clay soil that survive one year. If the probability that a tree planted in sandy soil will survive one year is 0.7 and the probability of one-year survival in clay soil is 0.6, compute an approximation to P(¡5 • X ¡ Y • 5). For the purposes of this exercise, ignore the continuity correction.

Vermont Technical College Electronic Applications ELT-2060 Lab 07: Common Mode Rejection Ratio and Instrumentation Amplifiers Reference: Laboratory Manual to Accompany Operational Amplifiers and Linear Integrated Circuits, Robert Coughlin, Sixth Edition. For the following exercise, make sure to record all calculations, estimations and measured results. Components: LM741 Op Amp, INA126 instrumentation Amp, 10 Ω, 1 kΩ, 470 Ω, 10 kΩ ½ W, 15 kΩ, 82 kΩ, 2-100 kΩ, , 50 kΩ potentiometer Objectives: a. LM741 Differential Voltage Gain, Common Mode Voltage Gain, Common Mode Rejection Ratio b. INA126 instrumentation amplifiers a. Differential Voltage Gain, Common Mode Voltage Gain, Common Mode Rejection Ratio Build the circuit of figure 1. Measure both E1 and E2 with respect to ground and record the values. Next calculate the differential voltage across the 10Ω resistor. Ediff= E1-E2. Calculate the differential voltage gain Adiff Vo = Adiff(E1-E2) or for this schematic Adiff = m = mR/R Measure Vo ` Modify the circuit as shown in figure 2, this now includes a common mode adjustment. In this circuit both inputs (+ input, – input) are shorted together and to E2, which is now the common mode voltage ECM. Measure and record E2 =ECM = __________ Measure Vo of the amplifier and adjust the 50k potentiometer for the smallest output voltage possible. Record this voltage as Vo-cm (Note this value should be approximately 1mV). Vo-cm=_________ Calculate common-mode voltage gain Acm = Vo-cm/Ecm Acm=_______ CMRR= Adiff/Acm CMRR =________ CMRR in dB = _________dB b. INA126 Instrumentation amplifiers 1. Wire the Instrumentation amplifier shown in figure 3. Set the differential gain Adiff to 10 by adjusting the 10-k Ω potentiometer. Again measure E1 and E2 with respect to ground E1=_____ E2=_______ 2. Predict the output voltage Vo from the equation Vo = Adiff(E1-E2) Calculated Vo= ____ Measured Vo=_____ 3. Readjust the 10-k Ω potentiometer for a differential gain of 100. Predict the output voltage Vo from the equation Vo = 100(E1-E2) Calculated Vo= ____ Measured Vo=_____ 4. To measure the common-mode voltage gain of the AD820 instrumentation amplifier, connect both inputs (pin 2 and pin 3) together and to E2 (see figure 4). Remeasure E2, in this configuration E2 = Ecm. Ecm=_______ 5. Measure Vocm Vocm=____ 6. Common-mode voltage gain = Acm= Vocm/Eocm 7. CMRR = Adiff/Acm CMRR = __________ CMRR in dB = _________ dB Compare your results to the INA126 data sheet Lab Report: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. Support your lab results using MultiSim; include your MultiSim schematics in your lab report to support your laboratory findings. Please include a written English language paragraph for all lab steps that required an explanation.

## Vermont Technical College Electronic Applications ELT-2060 Lab 07: Common Mode Rejection Ratio and Instrumentation Amplifiers Reference: Laboratory Manual to Accompany Operational Amplifiers and Linear Integrated Circuits, Robert Coughlin, Sixth Edition. For the following exercise, make sure to record all calculations, estimations and measured results. Components: LM741 Op Amp, INA126 instrumentation Amp, 10 Ω, 1 kΩ, 470 Ω, 10 kΩ ½ W, 15 kΩ, 82 kΩ, 2-100 kΩ, , 50 kΩ potentiometer Objectives: a. LM741 Differential Voltage Gain, Common Mode Voltage Gain, Common Mode Rejection Ratio b. INA126 instrumentation amplifiers a. Differential Voltage Gain, Common Mode Voltage Gain, Common Mode Rejection Ratio Build the circuit of figure 1. Measure both E1 and E2 with respect to ground and record the values. Next calculate the differential voltage across the 10Ω resistor. Ediff= E1-E2. Calculate the differential voltage gain Adiff Vo = Adiff(E1-E2) or for this schematic Adiff = m = mR/R Measure Vo ` Modify the circuit as shown in figure 2, this now includes a common mode adjustment. In this circuit both inputs (+ input, – input) are shorted together and to E2, which is now the common mode voltage ECM. Measure and record E2 =ECM = __________ Measure Vo of the amplifier and adjust the 50k potentiometer for the smallest output voltage possible. Record this voltage as Vo-cm (Note this value should be approximately 1mV). Vo-cm=_________ Calculate common-mode voltage gain Acm = Vo-cm/Ecm Acm=_______ CMRR= Adiff/Acm CMRR =________ CMRR in dB = _________dB b. INA126 Instrumentation amplifiers 1. Wire the Instrumentation amplifier shown in figure 3. Set the differential gain Adiff to 10 by adjusting the 10-k Ω potentiometer. Again measure E1 and E2 with respect to ground E1=_____ E2=_______ 2. Predict the output voltage Vo from the equation Vo = Adiff(E1-E2) Calculated Vo= ____ Measured Vo=_____ 3. Readjust the 10-k Ω potentiometer for a differential gain of 100. Predict the output voltage Vo from the equation Vo = 100(E1-E2) Calculated Vo= ____ Measured Vo=_____ 4. To measure the common-mode voltage gain of the AD820 instrumentation amplifier, connect both inputs (pin 2 and pin 3) together and to E2 (see figure 4). Remeasure E2, in this configuration E2 = Ecm. Ecm=_______ 5. Measure Vocm Vocm=____ 6. Common-mode voltage gain = Acm= Vocm/Eocm 7. CMRR = Adiff/Acm CMRR = __________ CMRR in dB = _________ dB Compare your results to the INA126 data sheet Lab Report: This lab requires a semi-formal lab report. Record all calculations, estimations, and measured results. Support your lab results using MultiSim; include your MultiSim schematics in your lab report to support your laboratory findings. Please include a written English language paragraph for all lab steps that required an explanation.

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What is meant by the term viscoelasticity

## What is meant by the term viscoelasticity

In actuality all materials depart from Hooke’s law in numerous … Read More...
Name Car Fuel Economy Objectives: 1. Collect data from a reliable source 2. Create a chart of important values 3. Determine what type of graph best fits the available data (exponential, linear, logarithmic, polynomial… etc) 4. Create a plot using the correct values and determine which variable goes on which axis 5. Fit a line to the graph and show equation 6. Use the equation to predict a future value 7. Use the equation editor to display equations Your task: You want to determine whether the curb weight of the vehicle has any impact on the Highway MPG of the vehicle. In order to do this, go to a reliable source and select 10 of your favorite cars. It can be from any make, model or year and be different across the board. As a matter of fact, the more variety you have in the make, model and year the better. (Please follow the correct homework heading format and please record your answers in blue color on this sheet) General note: Whenever you create a graph, label the X-axis and the Y-axis. Give the graph an appropriate title and include units for the axes. Use the Equation Editor to type answers as equations. 1. On Excel, create a table by recording the make, model and year of the vehicle, the curb weight, and the highway MPG (Miles per gallon). Do this for all 10 cars. 2. Create/insert a scatter plot of the Curb weight (X-axis) vs. Highway MPG (Y-axis). 3. Select the data points by right clicking on the data point and select to add a linear trendline for the 10 cars you currently have. Make it a solid green line (format the line). 4. Select to display the equation on the chart. What is the equation? _______________________________ 5. Using your equation, predict the MPG of a Uhaul that has a curb weight of 12,600lbs (type in curb weight as x). What is that MPG (y)? ___________________________. Record the Uhaul’s curb weight and predicted MPG on the excel sheet. Label clearly with the equation used. Does that answer make sense? Why or why not? ___________________________________________________ ___________________________________________________________________________________________ 6. According to the Uhaul website, a typical truck will get about 10 MPG. Create another scatter plot (do not delete the original one) that shows the same data, but is fitted with an exponential trendline (dashed red line). 7. Select to display the equation on the chart. What is the equation? ________________________________ 8. Using your equation, predict the MPG of a Uhaul that has a curb weight of 12,600lbs. What is that MPG? ___________________________. Record the Uhaul’s curb weight and predicted MPG on the excel sheet. Label clearly with the equation used. Does that answer make sense? Why or why not? Is this prediction better or worse than the first equation? ___________________________________________________________________________________________ ___________________________________________________________________________________________ 9. Now create a third scatter plot of curb weight and highway MPG. Add both a linear and exponential trendline. Make the linear trendline a solid green line and the exponential trendline a dashed red line (just like the two previous graphs). 10. Include a legend for the third graph. 11. Organize the excel sheet to make it look professional. 12. Please upload both this document with all of your answers and the Excel sheet that you created in the assignment on eCampus.

## Name Car Fuel Economy Objectives: 1. Collect data from a reliable source 2. Create a chart of important values 3. Determine what type of graph best fits the available data (exponential, linear, logarithmic, polynomial… etc) 4. Create a plot using the correct values and determine which variable goes on which axis 5. Fit a line to the graph and show equation 6. Use the equation to predict a future value 7. Use the equation editor to display equations Your task: You want to determine whether the curb weight of the vehicle has any impact on the Highway MPG of the vehicle. In order to do this, go to a reliable source and select 10 of your favorite cars. It can be from any make, model or year and be different across the board. As a matter of fact, the more variety you have in the make, model and year the better. (Please follow the correct homework heading format and please record your answers in blue color on this sheet) General note: Whenever you create a graph, label the X-axis and the Y-axis. Give the graph an appropriate title and include units for the axes. Use the Equation Editor to type answers as equations. 1. On Excel, create a table by recording the make, model and year of the vehicle, the curb weight, and the highway MPG (Miles per gallon). Do this for all 10 cars. 2. Create/insert a scatter plot of the Curb weight (X-axis) vs. Highway MPG (Y-axis). 3. Select the data points by right clicking on the data point and select to add a linear trendline for the 10 cars you currently have. Make it a solid green line (format the line). 4. Select to display the equation on the chart. What is the equation? _______________________________ 5. Using your equation, predict the MPG of a Uhaul that has a curb weight of 12,600lbs (type in curb weight as x). What is that MPG (y)? ___________________________. Record the Uhaul’s curb weight and predicted MPG on the excel sheet. Label clearly with the equation used. Does that answer make sense? Why or why not? ___________________________________________________ ___________________________________________________________________________________________ 6. According to the Uhaul website, a typical truck will get about 10 MPG. Create another scatter plot (do not delete the original one) that shows the same data, but is fitted with an exponential trendline (dashed red line). 7. Select to display the equation on the chart. What is the equation? ________________________________ 8. Using your equation, predict the MPG of a Uhaul that has a curb weight of 12,600lbs. What is that MPG? ___________________________. Record the Uhaul’s curb weight and predicted MPG on the excel sheet. Label clearly with the equation used. Does that answer make sense? Why or why not? Is this prediction better or worse than the first equation? ___________________________________________________________________________________________ ___________________________________________________________________________________________ 9. Now create a third scatter plot of curb weight and highway MPG. Add both a linear and exponential trendline. Make the linear trendline a solid green line and the exponential trendline a dashed red line (just like the two previous graphs). 10. Include a legend for the third graph. 11. Organize the excel sheet to make it look professional. 12. Please upload both this document with all of your answers and the Excel sheet that you created in the assignment on eCampus.

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University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Exam 1 26 April 2013 Name: Problem 1 (25 points) Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Answer the following questions: a. Find cov(i; ^ 1). b. If 0 = 4 what is the least squares estimate of 1? c. What is the variance of the estimate of part (b)? d. Is the estimate of part (b) unbiased? Problem 2 (25 points) Answer the following questions: a. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ^ 0 is BLUE (it has the smallest variance among all the linear unbiased estimators of 0). b. Consider the model of part (a). Find cov(ei; ^ Yi). c. Consider the simple regression model through the origin yi = 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that Pn i=1 xiei where ei = Yi ? ^ Yi. d. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2xi, and cov(i; j) = 0. Also, x is nonrandom. Is the assumption of constant variance satis ed in the following model? Please explain. Yi p xi = 0 p xi + 1 xi p xi + i p xi : Problem 3 (25 points): Answer the following questions: a. Consider the model yi = 0 + 1xi +i. Assume that E(i) = 0, var(i) = 2, and cov(i; j) = 0. Suppose we rescale the x values as x = x ? , and we want to t the model yi =  0 +  1xi + i. Find the least squares estimates of  0 and  1 . b. Refer to the model yi =  0 +  1xi +i of part (a). Find the SSE of this model and compare it to the SSE of the model yi = 0 + 1xi + i. What is your conclusion? c. Consider the simple regression model yi = 0 + 1xi + i, with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ESY Y = (n ? 1)2 + 2 1SXX, where SY Y = Pn i=1(yi ? y)2 and SXX = Pn i=1(xi ? x)2. d. Refer to the model of part (c). Find cov(i; ei). Problem 4 (25 points) Suppose that a simple linear regression of miles per gallon (Y ) on car weight (x) has been performed on 32 observations. The least squares estimates are ^ 0 = 68:17 and ^ 1 = ?1:112, with se = 4:281. Other useful information: x = 30:91 and P32 i=1(xi ? x)2 = 2054:8. Answer the following questions: a. Construct a 95% con dence interval for 1. b. Construct a 95% con dence interval for 2. c. What is the value of R2? d. Construct a con dence interval for 3 0 ? 2 1 ? 50.

## University of California, Los Angeles Department of Statistics Statistics 100C Instructor: Nicolas Christou Exam 1 26 April 2013 Name: Problem 1 (25 points) Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Answer the following questions: a. Find cov(i; ^ 1). b. If 0 = 4 what is the least squares estimate of 1? c. What is the variance of the estimate of part (b)? d. Is the estimate of part (b) unbiased? Problem 2 (25 points) Answer the following questions: a. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ^ 0 is BLUE (it has the smallest variance among all the linear unbiased estimators of 0). b. Consider the model of part (a). Find cov(ei; ^ Yi). c. Consider the simple regression model through the origin yi = 1xi + i with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that Pn i=1 xiei where ei = Yi ? ^ Yi. d. Consider the simple regression model yi = 0 + 1xi + i with E(i) = 0, var(i) = 2xi, and cov(i; j) = 0. Also, x is nonrandom. Is the assumption of constant variance satis ed in the following model? Please explain. Yi p xi = 0 p xi + 1 xi p xi + i p xi : Problem 3 (25 points): Answer the following questions: a. Consider the model yi = 0 + 1xi +i. Assume that E(i) = 0, var(i) = 2, and cov(i; j) = 0. Suppose we rescale the x values as x = x ? , and we want to t the model yi =  0 +  1xi + i. Find the least squares estimates of  0 and  1 . b. Refer to the model yi =  0 +  1xi +i of part (a). Find the SSE of this model and compare it to the SSE of the model yi = 0 + 1xi + i. What is your conclusion? c. Consider the simple regression model yi = 0 + 1xi + i, with E(i) = 0, var(i) = 2, and cov(i; j) = 0. Show that ESY Y = (n ? 1)2 + 2 1SXX, where SY Y = Pn i=1(yi ? y)2 and SXX = Pn i=1(xi ? x)2. d. Refer to the model of part (c). Find cov(i; ei). Problem 4 (25 points) Suppose that a simple linear regression of miles per gallon (Y ) on car weight (x) has been performed on 32 observations. The least squares estimates are ^ 0 = 68:17 and ^ 1 = ?1:112, with se = 4:281. Other useful information: x = 30:91 and P32 i=1(xi ? x)2 = 2054:8. Answer the following questions: a. Construct a 95% con dence interval for 1. b. Construct a 95% con dence interval for 2. c. What is the value of R2? d. Construct a con dence interval for 3 0 ? 2 1 ? 50.

University of California, Los Angeles Department of Statistics Statistics 100C … Read More...
Q1 A machining center in a local manufacturing company has five jobs to complete. The jobs are labeled 1, 2, 3, 4, and 5 based on the order they entered the shop. Your manager has asked you to compare two different job sequences to determine the best order for processing the jobs – First-In, First-Out (FIFO) and Shortest Processing Time (SPT). Compare the two sequences. Which sequence would you recommend and why? Be sure to include a variety of measures (average completion time, lateness, etc.) to compare the two sequencing rules. Include all calculations in your response q2 You work for a healthcare insurance company as an industrial engineer. As part of the company’s customer service department, there is a call center that responds to customer questions by phone. Your manager has asked you to perform an analysis of some data that the call center has collected over the last two years in an effort to determine how many workers are needed. In particular your manager wants to find out if there is a relationship between how long it takes to answer a call (independent variable) and whether or not customers will hang up (dependent variable). Is there a linear relationship between these two variables? Include a graph and analysis to support your opinion. If the goal of the call center is to have fewer than 15% of calls abandoned, how quickly must the call center respond? Learning Outcome #2. Data for Question 3 Average Answer Speed = Average number of seconds to answer an incoming call during the week % abandoned = % of calls that are abandoned (hang up) before being answered during the week. let me know if you can do excel work so I can send you the rest of the information and date

## Q1 A machining center in a local manufacturing company has five jobs to complete. The jobs are labeled 1, 2, 3, 4, and 5 based on the order they entered the shop. Your manager has asked you to compare two different job sequences to determine the best order for processing the jobs – First-In, First-Out (FIFO) and Shortest Processing Time (SPT). Compare the two sequences. Which sequence would you recommend and why? Be sure to include a variety of measures (average completion time, lateness, etc.) to compare the two sequencing rules. Include all calculations in your response q2 You work for a healthcare insurance company as an industrial engineer. As part of the company’s customer service department, there is a call center that responds to customer questions by phone. Your manager has asked you to perform an analysis of some data that the call center has collected over the last two years in an effort to determine how many workers are needed. In particular your manager wants to find out if there is a relationship between how long it takes to answer a call (independent variable) and whether or not customers will hang up (dependent variable). Is there a linear relationship between these two variables? Include a graph and analysis to support your opinion. If the goal of the call center is to have fewer than 15% of calls abandoned, how quickly must the call center respond? Learning Outcome #2. Data for Question 3 Average Answer Speed = Average number of seconds to answer an incoming call during the week % abandoned = % of calls that are abandoned (hang up) before being answered during the week. let me know if you can do excel work so I can send you the rest of the information and date

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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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For Day 24 Homework Cover Sheet Name:_________________________________________________ 1. Read Pages from 364-371, or watch the videos listed below  Percentage, Ratio and Proportions Problems (11 min) http://www.youtube.com/watch?v=oLoRCRXTYv4  Direct and Inverse Variation (5 min) http://www.youtube.com/watch?v=8x0rZklxLLE 2. Attempt problems from page 111 to 113 3. Write a summary and answer questions below from your reading or watching of the videos. a) What is a proportion? b) What is a direct variation? c) What is an inverse variation? 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 Solving a linear equation with several occurrences of the variable: Fractional forms with binomial numerators Translating a sentence into a compound inequality U.S. Customary area unit conversion with whole number values U.S. Customary unit conversion with whole number values U.S. Customary unit conversion with whole number values: Two-step conversion Using two steps to solve an equation with whole numbers Word problem involving multiple rates Word problem on combined variation Word problem on direct variation Word problem on inverse proportions Word problem on inverse variation Word problem on mixed number proportions Word problem on proportions: Problem type 1 Word problem on proportions: Problem type 2 Word problem with linear inequalities: Problem type 1 Word problem with linear inequalities: Problem type 2 Writing a direct variation equation Writing an equation that models variation Areas of rectangles with the same perimeter Circumference ratios Conversions involving measurements in feet and inches Finding an angle measure for a triangle with an extended side Finding an angle measure of a triangle given two angles Finding angle measures of a right or isosceles triangle given angles with variables Finding simple interest without a calculator Finding the missing length in a figure Finding the original price given the sale price and percent discount Finding the percentage increase or decrease: Advanced Finding the perimeter or area of a rectangle given one of these values Finding the radius or the diameter of a circle given its circumference Finding the sale price without a calculator given the original price and percent discount Finding the value for a new score that will yield a given mean Identifying and naming congruent triangles Identifying direct variation equations Identifying direct variation from ordered pairs and writing equations Identifying properties used to solve a linear equation Identifying similar or congruent shapes on a grid Identifying solutions to a linear equation in one variable: Two-step equations Identifying solutions to a linear inequality in one variable Perimeter of a piecewise rectangular figure Sides of polygons having the same perimeter Similar polygons Similar right triangles

## For Day 24 Homework Cover Sheet Name:_________________________________________________ 1. Read Pages from 364-371, or watch the videos listed below  Percentage, Ratio and Proportions Problems (11 min) http://www.youtube.com/watch?v=oLoRCRXTYv4  Direct and Inverse Variation (5 min) http://www.youtube.com/watch?v=8x0rZklxLLE 2. Attempt problems from page 111 to 113 3. Write a summary and answer questions below from your reading or watching of the videos. a) What is a proportion? b) What is a direct variation? c) What is an inverse variation? 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 Solving a linear equation with several occurrences of the variable: Fractional forms with binomial numerators Translating a sentence into a compound inequality U.S. Customary area unit conversion with whole number values U.S. Customary unit conversion with whole number values U.S. Customary unit conversion with whole number values: Two-step conversion Using two steps to solve an equation with whole numbers Word problem involving multiple rates Word problem on combined variation Word problem on direct variation Word problem on inverse proportions Word problem on inverse variation Word problem on mixed number proportions Word problem on proportions: Problem type 1 Word problem on proportions: Problem type 2 Word problem with linear inequalities: Problem type 1 Word problem with linear inequalities: Problem type 2 Writing a direct variation equation Writing an equation that models variation Areas of rectangles with the same perimeter Circumference ratios Conversions involving measurements in feet and inches Finding an angle measure for a triangle with an extended side Finding an angle measure of a triangle given two angles Finding angle measures of a right or isosceles triangle given angles with variables Finding simple interest without a calculator Finding the missing length in a figure Finding the original price given the sale price and percent discount Finding the percentage increase or decrease: Advanced Finding the perimeter or area of a rectangle given one of these values Finding the radius or the diameter of a circle given its circumference Finding the sale price without a calculator given the original price and percent discount Finding the value for a new score that will yield a given mean Identifying and naming congruent triangles Identifying direct variation equations Identifying direct variation from ordered pairs and writing equations Identifying properties used to solve a linear equation Identifying similar or congruent shapes on a grid Identifying solutions to a linear equation in one variable: Two-step equations Identifying solutions to a linear inequality in one variable Perimeter of a piecewise rectangular figure Sides of polygons having the same perimeter Similar polygons Similar right triangles

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