Question

Gymnast Clothing manufactures expensive hockey jerseys for sale to college bookstores in runs of up to 300. Its cost (in dollars) for a run of x hockey jerseys is

C(x) = 3000 + 10x + 0.2x2 (0 ≤ x ≤ 300)
Gymnast Clothing sells the jerseys at $110 each. Find the revenue function.
R(x) =


Find the profit function.
P(x) =


How many should Gymnast Clothing manufacture to make a profit? HINT [See Example 2.] (Round your answer up to the nearest whole number.)

Answer 1

The revenue function for Gymnast Clothing is R(x) = 110x. The profit function is P(x) = 110x - (3000 + 10x + 0.2x^2). To determine the profit maximizing quantity, we need to find the value of x that makes P(x) greater than zero.

Step-by-step explanation:

To find the revenue function for Gymnast Clothing, we need to multiply the number of hockey jerseys sold (x) by the selling price per jersey ($110). Therefore, the revenue function is R(x) = 110x.

To find the profit function, we subtract the cost function C(x) from the revenue function R(x). Therefore, the profit function is P(x) = R(x) - C(x). Substituting the given cost function, we have P(x) = 110x - (3000 + 10x + 0.2x^2).

To determine how many jerseys Gymnast Clothing should manufacture to make a profit, we need to find the value of x that makes P(x) greater than zero. We can solve this mathematically or graphically. In this case, it is more efficient to solve it graphically by plotting the profit function and finding the quantity that corresponds to a positive profit.

Answer 2

Given that Gymnast Clothing sells the jerseys at $110 each, the revenue function is given by


`R(x)=110x`

Profit function is given by revenue function minus cost function, i.e.


`P(x)=R(x)-C(x)=110x-(</span><span>3000+10x+0.2x^2)=110x-3000-10x-0.2x^2=-0.2x^2+100x-3000`

Thus,

`P(x)=-0.2x^2+100x-3000`

The number Gymnast Clothing should manufacture to make a profit is given by

`-0.2x^2+100x-3000=0 \\ \\ \Rightarrow0.2x^2-100x+3,000=0 \\ \\ \Rightarrow x=468, \ 32`

Because, Gymnast Clothing manufactures expensive hockey jerseys for sale to college bookstores in runs of up to 300.

The number Gymnast Clothing should manufacture to make a profit is 32.