Question

A t-shirt shop determines that the profit function for selling each t-shirt at price

p is given by the function
f(p) = -20p2 + 700p - 2800.
Determine the maximum
profit and the price charged to reach this maximum profit.
a) Maximum profit: $5600, Price: $17.50
b) Maximum profit: $4000, Price: $17.50
c) Maximum profit: $5600, Price: $17.50
d) Maximum profit: $4000, Price: $15.00

Answer 1

The maximum profit for the t-shirt shop is $5600, and the price charged to reach this maximum profit is $17.50.

Step-by-step explanation:

The profit function for selling each t-shirt at price p is given by the function f(p) = -20p2 + 700p - 2800. To determine the maximum profit, we need to find the vertex of the quadratic function.

The vertex of a quadratic function of the form f(x) = ax2 + bx + c is given by the formula:

x = -b / (2a)

In this case, a = -20 and b = 700. Plugging in these values into the formula, we get:

x = -700 / (2(-20)) = 17.5

So, the price charged to reach the maximum profit is $17.50.

Now, to find the maximum profit, we substitute the value of x into the profit function:

f (17.5) = -20(17.5)2 + 700(17.5) - 2800 = $5600

Therefore, the maximum profit is $5600.