Question

a snack food company uses the equation $p(n) = -4n^{2} + 25n - 36$ to model the profit it makes, in thousands of dollars, when it sells $n$ million bags of gourmet chips. the company will break even when its profit is 0. complete the sentences. write your answers as whole or decimal numbers. do not round. the company will break even when it sells \boxed{} million or \boxed{} million bags of gourmet chips. between those points, the company will be profitable.

Explanation:

Step1: Set profit to zero

To find when the company breaks even, we set \( P(n) = 0 \), so we have the equation:
\[
-4n^2 + 25n - 36 = 0
\]
Multiply both sides by -1 to make the quadratic coefficient positive:
\[
4n^2 - 25n + 36 = 0
\]

Step2: Solve the quadratic equation

We use the quadratic formula \( n=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for a quadratic equation \( ax^{2}+bx + c = 0 \). Here, \( a = 4 \), \( b=-25 \), and \( c = 36 \).

First, calculate the discriminant \( \Delta=b^{2}-4ac \):
\[
\Delta=(-25)^{2}-4\times4\times36=625 - 576 = 49
\]

Then, find the roots:
\[
n=\frac{25\pm\sqrt{49}}{2\times4}=\frac{25\pm7}{8}
\]

Step3: Calculate the two roots

For the plus sign:
\[
n=\frac{25 + 7}{8}=\frac{32}{8}=4
\]

For the minus sign:
\[
n=\frac{25-7}{8}=\frac{18}{8}=2.25
\]

Answer:

The company will break even when it sells \( 2.25 \) million or \( 4 \) million bags of gourmet chips.