Question

Suppose the Sunglasses Hut Company has a profit function given by P(q)=−0.02q2+4q−20P(q)=-0.02q2+4q-20, where qq is the number of thousands of pairs of sunglasses sold and produced, and P(q)P(q) is the total profit, in thousands of dollars, from selling and producing qq pairs of sunglasses.
A) How many pairs of sunglasses (in thousands) should be sold to maximize profits? (If necessary, round your answer to three decimal places.)
B) What are the actual maximum profits (in thousands) that can be expected? (If necessary, round your answer to three decimal places.)

Answer 1

Answer:

(A) 100 (in thousands)

(B) 180 (in thousand dollars)

Step-by-step explanation:

Given:

profit function as P(q) = -0.02q² + 4q - 20

where,

q is the number of thousands of pairs of sunglasses sold and produced,

P(q) is the total profit in thousands of dollars

To find the point of maxima differentiating the above equation and equating it to zero

P'(q) = - (2)0.02q + 4 - 0 = 0

or

⇒ - 0.04q + 4 = 0

or

⇒ - 0.04q = - 4

or

⇒ q = 100

Hence,

(A) 100 pairs of sunglasses (in thousands) should be sold to maximize profits

(B) Substituting the value of q in the profit function to calculate the actual maximum profit

P(q) = -0.02(100)² + 4(100) - 20

or

P(q) = - 0.02(10000) + 400 - 20

or

P(q) = - 0.02(10000) + 400 - 20

or

P(q) = 180 (in thousands dollar)