Q A screen printer produces custom silkscreen apparel The pr

Q:

A screen printer produces custom silk-screen apparel. The profit P(x) from the sale of x T-shirts in dollars is given by P(x) = -0.05x² + 7.75x - 200

Determine the number of T-shirts that should be produced to result in the maximum profit and find the maximum profit. (Note: Only whole T-shirts are produced.)

Solution

P(x) = -0.05x^2 + 7.75x - 200

P\'(x) = 0 to maximize the profit function.

Hence, -0.05*2x + 7.75 = 0

x = 7.75/(0.05*2) = 77.5

since, no. of tshirts should be whole no. So, we will put x=77 and x=78,

P(77) = -(0.05) * (77)^2 + 7.75 * 77 - 200 = 100.3

P(78) = -(0.05) * (78)^2 + 7.75 * 78 - 200 = 100.3

So, both 77 tshirts and 78 tshirts are possible for maximizing profit.