Two fraternities, Sig Ep and Ep Sig, plan to raise money jointly to benefit homeless people on Long Island. They will sell Yoda vs. Alien T-shirts in the student center, but are not sure how much to charge. Sig Ep treasurer Augustus recalls that they once sold 70 shirts in a week at $9 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 420 per week at $4 per shirt.
(a) Based on this information, construct a linear demand equation for Yoda vs. Alien T-shirts, and hence obtain the weekly revenue R as a function of the unit price x.
R(x) =
.
(b) The university administration charges the fraternities a weekly fee of $200 for use of the Student Center. Find the weekly profit P as a function of the unit price x. HINT [See Example 4.]
P(x) =
.
Determine how much the fraternities should charge to obtain the largest possible weekly profit.
x = $ . per T-shirt
What is the largest possible weekly profit?
$
To construct a linear demand equation, we can use the given information from both treasurers.
(a) Let's start by finding the slope of the demand equation. The slope represents the change in the number of shirts sold per $1 change in price. We can calculate the slope using the following formula:
Slope (m) = (change in quantity)/(change in price)
Using the information given:
For Sig Ep, the change in quantity is 70 shirts, and the change in price is $9 - $4 = $5.
So, the slope for Sig Ep is 70/$5 = 14 shirts per dollar.
For Ep Sig, the change in quantity is 420 shirts, and the change in price is $4 - $9 = -$5 (since the price decreased).
So, the slope for Ep Sig is 420/(-$5) = -84 shirts per dollar.
Now let's find the y-intercept, which represents the quantity demanded when the price is $0. We can use either fraternity's data to find the y-intercept.
Using Sig Ep's data:
When the price is $9, Sig Ep sells 70 shirts.
So, the y-intercept is (0, 70).
Now that we have the slope and y-intercept, we can construct the linear demand equation using the slope-intercept form: y = mx + b.
For Sig Ep: y = 14x + 70
For Ep Sig: y = -84x + 70
The weekly revenue R can be obtained by multiplying the unit price x by the quantity demanded (given by the demand equation):
For Sig Ep: R(x) = x * (14x + 70) = 14x^2 + 70x
For Ep Sig: R(x) = x * (-84x + 70) = -84x^2 + 70x
(b) To find the weekly profit P, we need to deduct the weekly fee of $200 from the revenue R. So we have:
For Sig Ep: P(x) = R(x) - $200 = 14x^2 + 70x - $200
For Ep Sig: P(x) = R(x) - $200 = -84x^2 + 70x - $200
To determine the price that will result in the largest possible weekly profit, we need to find the maximum value of P(x). We can do this by finding the vertex of the quadratic function (14x^2 + 70x - $200) and (84x^2 + 70x - $200).
Using the formula x = -b/2a, where a and b are the coefficients of x^2 and x respectively:
For Sig Ep: x = -70/(2*14) = -70/28 = -2.5
For Ep Sig: x = -70/(2*-84) = -70/-168 = 0.417
Since prices cannot be negative, the fraternities should charge $2.50 per T-shirt for Sig Ep and $0.42 per T-shirt for Ep Sig to obtain the largest possible weekly profit.
To find the largest possible weekly profit, substitute the respective values of x into the profit equation:
For Sig Ep: P($2.50) = 14($2.50)^2 + 70($2.50) - $200 = $275
For Ep Sig: P($0.42) = -84($0.42)^2 + 70($0.42) - $200 = $28.924
The largest possible weekly profit would be $275 for Sig Ep and $28.924 for Ep Sig.