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) = 2750 + 8x + 0.1x2 (0 ≤ x ≤ 500). Gymnast Clothing sells the cleats at $100 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?

Answer 1

Revenue function:
`R(x)=100x`

Profit function:
`P(x)=-0.1 x^2 + 92 x - 2750`

31 to 500 Gymnast Clothing are manufactured to make a profit.

Explanation:

The cost (in dollars) for a run of x pairs of cleats.

`C(x)=2750 + 8x + 0.1x^2`

Gymnast Clothing sells the cleats at $100 per pair.

The Revenue (in dollars) for a run of x pairs of cleats.

`R(x)=100x`

Profit = Revenue - Cost

`P(x)=R(x)-C(x)`

`P(x)=100x-(2750 + 8x + 0.1x^2)`

`P(x)=-0.1 x^2 + 92 x - 2750`

We need to find how many should Gymnast Clothing manufacture to make a profit.

`P(x)\geq 0`

`-0.1 x^2 + 92 x - 2750\geq 0`

From the given figure it is clear that P(x) is greater than or equal to 0 for 30.931 ≤ x ≤ 889.069.

The value of x can not be more than 500.

31 ≤ x ≤ 500

31 to 500 Gymnast Clothing are manufactured to make a profit.