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?
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.
