Question

The function P(t) = -5t2 + 70t + 600 models a company's profit in thousands of dollars, where t is the

number of years since 1990.

a) In what year will the company reach its maximum profit?

b) What is the company's max profit?

c) How much money will the company have in 2002²

Answer 1

a) The maximum profit occurs at the vertex of the parabola, which has an x-value of -b/2a. In this case, the function is P(t) = -5t^2 + 70t + 600, so a = -5 and b = 70. Therefore, the vertex occurs at t = -b/2a = -70/(2*(-5)) = 7. The number 7 represents the number of years after 1990, so the company will reach its maximum profit in 1997.

b) To find the maximum profit, we need to evaluate the function at the x-value of the vertex. In this case, the vertex occurs at t = 7, so the maximum profit is P(7) = -5(7)^2 + 70(7) + 600 = $745,000.

c) To find the profit in 2002^2, we need to find the value of t that corresponds to the year 2002^2. 2002^2 = 4008004, which is 14 years after 1990. Therefore, we need to find P(14). P(t) = -5t^2 + 70t + 600, so P(14) = -5(14)^2 + 70(14) + 600 = $680.