Selling shirts. If a vendor charges p dollars each for
rugby shirts, then he expects to sell 2000 � 100p shirts at
a tournament.
a) Find a polynomial R(p) that represents the total revenue
when the shirts are p dollars each.
b) Find R(5), R(10), and R(20).
c) Use the bar graph to determine the price that will give
the maximum total revenue.
To find the polynomial R(p) that represents the total revenue when the shirts are p dollars each, we need to multiply the price per shirt, p, by the number of shirts sold, which is given as 2000 - 100p. So the equation for the total revenue is R(p) = p * (2000 - 100p).
To find R(5), we substitute p = 5 into the equation R(p) = p * (2000 - 100p).
R(5) = 5 * (2000 - 100 * 5) = 5 * (2000 - 500) = 5 * 1500 = 7500.
To find R(10), we substitute p = 10 into the equation R(p) = p * (2000 - 100p).
R(10) = 10 * (2000 - 100 * 10) = 10 * (2000 - 1000) = 10 * 1000 = 10000.
To find R(20), we substitute p = 20 into the equation R(p) = p * (2000 - 100p).
R(20) = 20 * (2000 - 100 * 20) = 20 * (2000 - 2000) = 20 * 0 = 0.
Next, to determine the price that will give the maximum total revenue, we can create a bar graph with the price per shirt (p) on the horizontal axis and the total revenue (R(p)) on the vertical axis. Plot the values of R(p) for different values of p and observe the highest point on the graph. The price corresponding to that highest point will give the maximum total revenue.
Now, you have the polynomial representation R(p) = p * (2000 - 100p), and the values R(5) = 7500, R(10) = 10000, and R(20) = 0. Plot these points on a graph and identify the price that corresponds to the highest point on the graph. This price will give the maximum total revenue.