Question

Emilio's Publishing 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.

Graph of function p of x equals negative 0.1 x squared plus 15 x plus 120. The graph has the x-axis labeled as number of price increases, and the y-axis labeled as profit. The curve begins at (0, 120), increases to (75, 682.5), and decreases through (157.614, 0).

Approximately what amount should the company sell each cookbook for to maximize profit? (1 point)
$8.25
$9.50
$10.20
$17.00

Answer 1

Using the vertex of the quadratic equation, it is found that the profit will be maximized when each cookbook is sold for $8.25.

What is the vertex of a quadratic equation?

A quadratic equation is modeled by:

$y = ax^2 + bx + c$

The vertex is given by:

$(x_v, y_v)$

In which:

$x_v = -\frac{b}{2a}$

$y_v = -\frac{b^2 - 4ac}{4a}$

Considering the coefficient a, we have that:

• If a < 0, the vertex is a maximum point.

• If a > 0, the vertex is a minimum point.

In this problem, the function is modeled as follows, considering x as the price divided by $0.05.

P(x) = -0.1x² + 15x + 120.

The coefficients are a = -0.1 < 0, b = 15, c = 120, hence the x-value of the vertex is given by:

$x_v = -\frac{15}{2(-0.1)} = \frac{15}{0.2} = 75$

75 increases of $0.05, hence the price will be of:

4.5 + 75 x 0.05 = $8.25.

Hence, the profit will be maximized when each cookbook is sold for $8.25.

More can be learned about the vertex of a quadratic equation at https://brainly.com/question/24737967