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

Answer:

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.

Answer 2

Revenue, R = 130x

Profit, P = Revenue – Cost

 P = R – C

P = 130x – 2950 + 6x + 0.1x^2

At P = 0

0 = -2950 + 136x +0.1x^2

X = 21.355, -1381.355

Therefore, manufacture greater than 21.355 items to make profit.