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) = 3200 + 9x + 0.1x2 (0 ≤ x ≤ 500). gymnast clothing sells the cleats at $150 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 + 0.1x²)

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.