Question

100 points please be quick but thorough

The profit P (in dollars) made by a cinema from selling x bags of popcorn can be modeled by
P = 2.38x − ((x^2)/( 20,000)) − 3,300, 0 ≤ x ≤ 50,000.

1. Find the intervals on which P is increasing and decreasing. (Enter your answers using interval notation.)
Increasing_______
Decreasing_______


If you owned the cinema, what price would you charge to obtain a maximum profit for popcorn? Explain your reasoning.

Options:
a) You should charge the price that yields sales of x = 23,800 bags of popcorn. Since the function changes from increasing to decreasing at x = 23,800, the maximum profit occurs at this value.
b) You should charge the price that yields sales of x = 24,300 bags of popcorn. Since the function changes from decreasing to increasing at x = 24,300, the maximum profit occurs at this value.
c) You should charge the price that yields sales of x = 23,700 bags of popcorn. Since the function changes from decreasing to increasing at x = 23,700, the minimum profit occurs at this value.
d) You should charge the price that yields sales of x = 23,800 bags of popcorn. Since the function changes from decreasing to increasing at x = 23,800, the minimum profit occurs at this value.
e) You should charge the price that yields sales of x = 23,300 bags of popcorn. Since the function changes from increasing to decreasing at x = 23,300, the minimum profit occurs at this value.

Answer 1

increasing: (0, 23,800)

decreasing: (23,800, 50,000)

To obtain maximum profit for popcorn, you should charge the price that yields sales of x = 23,800 bags of popcorn. Since the function changes from increasing to decreasing at x = 23,800, the maximum profit occurs at this value. Therefore, the correct option is a) You should charge the price that yields sales of x = 23,800 bags of popcorn.