Number Who Chose This Maximum Amount

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1 1 TASK 3.3.1: MAXIMIZING REVENUE AND PROFIT Solutions Your school is trying to oost interest in its athletic program. It has decided to sell a pass that will allow the holder to attend all athletic events at the school. The 800 families in the community were surveyed and asked the question, What is the most you would pay for an all-sports season pass? [It was assumed that if you were willing to pay a given amount, you would also e willing to pay less than that.] Here are the survey results. Maximum Price of Season Pass Numer Who Chose This Maximum Amount Numer Willing to Pay this Price Revenue 1. How can you determine the revenue at each price? To determine the revenue at each price, multiply the price column y the numer of willing uyers at this cost. 2. Lael the third column in the tale aove revenue and fill in the tale. Then make a scatter plot of the revenue versus the price. Sketch the scatter plot elow. Below is the tale with the completed revenue column in L 3 and the scatter plot. 3. Find a quadratic equation that fits the data in the scatter plot. Many participants choose to approach this question y performing a quadratic regression on L 1 and L 3. A regression equation is y =! x x!

2 2 Ask participants to consider an alternate method that uses matrices. As a group, go through the Transparency Using Matrices to Solve a System of Equations. Use the following questions to guide the discussion. What is the general form of a quadratic equation? y = ax 2 + x + c What do the data points in L 1 and L 3 represent in this equation? The values in L 1 represent the x-values and the values in L 3 represent the y-values. Choose a data point and sustitute the values into the general quadratic equation. For example, choose (50, 38250). Then, = a 50 2 ( ) What do we need in order to solve for a,, and c? We need three equations like the one aove so that we can solve a system of three equations for three unknowns. Choose two other points and set up a system of three equations to solve for the unknowns a,, and c. For example, choose (100, 49500) and (135, 39150). The system would e: = a = a = a ( ) + 50 ( ) ( ) How can the system aove e rewritten as a matrix equation?! ! a! ' = " % " c% " 39150% Notice in the matrix equation aove, we have a 3 y 3 matrix with x-values multiplied y the coefficient matrix to give us the resultant y-value matrix. Remind participants of the row y column matrix multiplication to see that this matrix equation is indeed equivalent to the system of three equations. Next, have participants store the x-value matrix into matrix [A] on their calculators and the y-value matrix into matrix [B]. The aove matrix equation is [A] [coefficients] = [B]. How can we solve for the coefficient matrix? Multiply oth sides of the equation y [A] -1. It is important to rememer at this point that matrix multiplication is not commutative. So, we must e precise in our multiplication. The solution is [coefficient] = [A] -1 [B]. This calculation can e performed quickly using a graphing calculator. The quadratic equation is y =!6.126x x! This gives the revenue function R(x).

3 3 Other examples: Selecting the points (50, 38250), (135, 39150) and (90, 48600) yields y =!5.5147x x Selecting the points (100, 49500), (135, 39150) and (150, 31500) yields y =!4.2857x x Have each group choose a different set of 3 points and use matrices to fit a quadratic function to the data. Compare the results. 4. Find the price of the season pass that produces the maximum revenue. The maximum revenue of 49, is otained when the price is This can e found y using the maximum feature on a graphing calculator once the equation is found. How does the maximum found using the calculator feature compare to what you might have guessed the maximum price should have een using only the survey results? Using the survey results, we might have overpriced the season pass. The maximum data point is a price of 100 resulting in a revenue of 49,500 as opposed to the actual maximum revenue of 49, The cost of the sporting events to the school is also a function of the price of a season pass. Suppose C x ( ) is a cost function such that C x ( ) =!150x + 54,000. When will the school make a profit? Descrie this in general terms as well as specifically for this situation. The school will make a profit when R(x) > C(x). The intersection points of R(x) and C(x) are called reak-even points. The profit function can e written as P(x)=R(x)- C(x). What kind of function is the profit function? It is also a quadratic ecause it is the result of a quadratic function minus a linear function. The school will make a profit if the price is set etween and Graphically this can e seen y looking at the intersection of the revenue and cost functions.

4 4 Another way to see when the school will make a profit is to look at the graph of P(x) and find where it is aove the x-axis. 6. What price would produce the maximum profit for the school? Is this the same price that would produce the maximum revenue? The price that would produce the maximum profit is This is not the same as the price that would maximize the revenue. Math notes Many participants choose to approach this question y performing a quadratic regression on L 1 and L 3. A regression equation is y =! x x! Ask participants to consider an alternate method that uses matrices. As a group, go through the Transparency Using Matrices to Solve a System of Equations. Use the following questions to guide the discussion. What is the general form of a quadratic equation? What do the data points in L 1 and L 3 represent in this equation? Choose a data point and sustitute the values into the general quadratic equation. What do we need in order to solve for a,, and c?

5 5 Choose two other points and set up a system of three equations to solve for the unknowns a,, and c. For example, choose (100, 49500) and (135, 39150). The system would e: = a = a = a ( ) + 50 ( ) ( ) How can the system aove e rewritten as a matrix equation?! ! a! ' = " % " c% " 39150% Notice in the matrix equation aove, we have a 3 y 3 matrix with x-values multiplied y the coefficient matrix to give us the resultant y-value matrix. Remind participants of the row y column matrix multiplication to see that this matrix equation is indeed equivalent to the system of three equations. Next, have participants store the x-value matrix into matrix [A] on their calculators and the y-value matrix into matrix [B]. The aove matrix equation is [A] [coefficients] = [B]. How can we solve for the coefficient matrix? Use the instructor s transparency to descrie how to solve for the coefficient matrix. Ask each group to choose a different set of 3 points and use matrices to fit a quadratic to the data. Compare the results. Teaching notes In this task, participants look at a quadratic situation and find an equation to fit the data. Go over the scenario given on the activity sheet. Have participants work in groups of 3 4 on Exercises 1 3.

6 6 TASK 3.3.1: MAXIMIZING REVENUE AND PROFIT Your school is trying to oost interest in its athletic program. It has decided to sell a pass that will allow the holder to attend all athletic events at the school. The 800 families in the community were surveyed and asked the question, What is the most you would pay for an all-sports season pass? [It was assumed that if you were willing to pay a given amount, you would also e willing to pay less than that.] Here are the survey results. Maximum Price of Season Pass Numer Who Chose This Maximum Amount Numer Willing to Pay this Price Revenue 1. How can you determine the revenue at each price? 2. Lael the third column in the tale aove revenue and fill in the tale. Then make a scatter plot of the revenue versus the price. Sketch the scatter plot elow. 3. Find a quadratic equation that fits the data in the scatter plot. 4. Find the price of the season pass that produces the maximum revenue.

7 7 5. The cost of the sporting events to the school is also a function of the price of season pass. Suppose C ( x) is a cost function such that C x When will the school make a profit? Descrie this in general terms as well as specifically for this situation. ( ) =!150x + 54, What price would produce the maximum profit for the school? Is this the same price that would produce the maximum revenue?

8 8 Instructor s Transparency Using Matrices to Solve a System of Equations 38250=a(50 2 ) + (50) + c 49500=a(100 2 ) + (100) + c 39150=a(135 2 ) + (135) + c Let Matrix A = 2! " % and Matrix B =! 38250" % Then the system aove can e written as: A! a" % c =B To solve for the coefficients: A -1 A! a" % c =A -1 B I! a" % c = A -1 B " a "! % = % % % c% '! % '