Ymnast 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) = 3250 + 5x + 0.1x2 (0 ≤ x ≤ 500).

Gymnast Clothing sells the cleats at $110 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?

The relationship between profit and revenue is:

Profit = Revenue - Total Cost

Since the cost is given as a function, the equation will be substituted as

Profit = Revenue - (3,250 + 5x + 0.1x ²)

Now, the revenue is a linear graph because it is the number of pairs multiplied the fixed rate. So, the slope is constant.

Revenue = 110x - - > this is the function for revenue

Thus, the equation for profit would be

Profit = 110x - 3,250 - 5x - 0.1x²

P = - 0.1x²+105x-3,250 - - > this is the function for profit

Now, to find the minimum value of x to make profit, differentiate P with respect to x and equate to 0.

dP/dx = 0 = - 0.2x + 105

x = 525

Thus, the Gymnast Clothing should manufacture at least 525 pairs in order to make profit.