Question

A local civic group is selling t-shirts to raise funds for Relay to Life. They plan to sell

2500 t-shirts for $10. They consider raising the t-shirt price in order to earn more
profit. For each $1 increase, they will sell 100 fewer t-shirts. Let x represent the
number of $1 increases.
a. Write a function,
Write a function, c(x), to represent the cost of one t-shirt based on the number
of increases.
b. Write a function, t(x), to represent the number of t-shirts sold based on the
number of increases.
c. Write a revenue function, r(x), for the t-shirt sell that could be used to maximize

Answer 1

a. Let us say that c(x) is the cost of one t shirt based on the number of increases therefore:

c = 10 + x

b. Say that t(x) is the number of t shirts sold based on the number of increases so:

t = 2500 – 100 x

b. The revenue is simply the product:

r(x) = (10 + x) (2500 – 100 x)

Combining:

r(x) = 25000 – 1000 x + 2500 x – 100 x^2

r(x) = 25000 + 1500 x – 100 x^2

Taking the 1st derivative and setting dr / dx = 0 to get the maxima:

dr / dx = 1500 – 200 x

1500 = 200 x

x = 7.5

t = 2500 – 100 x = 2500 – 100 (7.5) = 1750

Hence 1750 shirts should be sold.