Question

Gymnast Clothing manufactures expensive soccer cleats for sale to college bookstores in runs of up to 500. Its cost (in dollars) for a run of x pairs of cleats is: C(x) = 2950 + 6x + 0.1x2 (0 ≤ x ≤ 500). Gymnast Clothing sells the cleats at $130 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?

Answer 1

The revenue function for Gymnast Clothing is R(x) = 130x, and the profit function is P(x) = 130x - (2950 + 6x + 0.1x^2). To make a profit, Gymnast Clothing must produce and sell more cleats than the break-even point which is determined by setting the profit function equal to zero and solving for x.

Step-by-step explanation:

To find the revenue function, we need to multiply the number of cleats, x, by the price at which they are sold. Since Gymnast Clothing sells the cleats at $130 per pair, the revenue function R(x) is R(x) = 130x.

The profit function is obtained by subtracting the cost function from the revenue function. Therefore, the profit function P(x) is P(x) = R(x) - C(x) = 130x - (2950 + 6x + 0.1x2).

The company needs to produce and sell cleats in order to cover the fixed costs and variable costs. To calculate the break-even point where profit is zero, we can set P(x) = 0 and solve for x. The specific number of cleats to manufacture to ensure a profit should be greater than the break-even point.

Answer 2

Manufacture more than 24 items to make a profit.

Step-by-step explanation:

Let's find the Revenue function:

Revenue (
`R`) = Price per pair * Number of pairs sold

`R(x)=130x`

Now let's find the profit function:

Profit (
`P(x)`) = Revenue (
`R(x)`) - Cost
`C(x)`

(
`P(x)`) = (
`R(x)`) –
`C(x)`

`P(x)=130x-(2950+6x+0.1x^(2) )`, Where, (
`0\leq x\leq 500`)

The requirement to make a profit:

`P(x)\geq 0`

⇒
`130x-(2950+6x+0.1x^(2) )\geq 0`

`x=24.265` or x=1215.735

Therefore, they should manufacture more than 24 items to make a profit.