Question

A publishing company prints and sells cookbooks to stores for $4.50 each. Their weekly profits can be modeled by the quadratic function P(x) = -0.1x2 + 15x + 120, where P(x) is profit and x

is the number of $0.05 price increases. Use the graph to answer the question.

• The profit at 100 price increases is equal to the profit at 50 price increases.
O The profit at 0 price increases is equal to the profit at 150 price increases.
• The maximum profit occurs at 75 price increases.
• The maximum profit occurs at 158 price increases.

Answer 1

Based on the analysis, the correct statements are:

A. The profit at 100 price increases is equal to the profit at 50 price increases.

B. The profit at 0 price increases is equal to the profit at 150 price increases.

C. The maximum profit occurs at 75 price increases.

How to determine which statements are true

To determine which statements are true based on the given quadratic function
`P(x) = -0.1x^2 + 15x + 120`, analyze the properties of the function and compare them to the statements.

The profit at 100 price increases is equal to the profit at 50 price increases.

To verify this statement, compare the profit values at x = 100 and x = 50.

P(100) =
`-0.1(100)^2` + 15(100) + 120 = -1000 + 1500 + 120 = 620

P(50) =
`-0.1(50)^2` + 15(50) + 120 = -250 + 750 + 120 = 620

The profit at 100 price increases is indeed equal to the profit at 50 price increases. Therefore, this statement is true.

The profit at 0 price increases is equal to the profit at 150 price increases.

To verify this statement, compare the profit values at x = 0 and x = 150.

P(0) =
`-0.1(0)^2` + 15(0) + 120 = 120

P(150) =
`-0.1(150)^2` + 15(150) + 120 = -2250 + 2250 + 120 = 120

The profit at 0 price increases is indeed equal to the profit at 150 price increases. Therefore, this statement is true.

The maximum profit occurs at 75 price increases.

To find the maximum profit, determine the x-coordinate of the vertex of the quadratic function.

The vertex of a quadratic function is given by the formula x = -b / (2a), where a and b are the coefficients of the quadratic function.

For the function
`P(x) = -0.1x^2 + 15x + 120`, a = -0.1 and b = 15.

x = -15 / (2*(-0.1)) = -15 / (-0.2) = 75

The maximum profit occurs at x = 75 price increases. Therefore, this statement is true.

The maximum profit occurs at 158 price increases.

From the calculation in the previous step, we found that the maximum profit occurs at x = 75 price increases. Therefore, this statement is false.

Based on the analysis, the correct statements are:

The profit at 100 price increases is equal to the profit at 50 price increases.

The profit at 0 price increases is equal to the profit at 150 price increases.

The maximum profit occurs at 75 price increases.