Answer:
The least amount is $75.50
Step-by-step explanation:
Given
[tex]p \to profit[/tex]
[tex]x \to amount[/tex]
[tex]p = -0.5x^2 + 36x - 231[/tex]
Required
The smallest amount to make at least 363
We have:
[tex]p = -0.5x^2 + 36x - 231[/tex]
Rewrite as:
[tex]-0.5x^2 + 36x - 231=p[/tex]
Substitute 363 for p
[tex]-0.5x^2 + 36x - 231 = 363[/tex]
Collect like terms
[tex]-0.5x^2 + 36x - 231 + 363 = 0[/tex]
[tex]-0.5x^2 + 36x + 132 = 0[/tex]
Using quadratic formula, we have:
[tex]x = \frac{-b \± \sqrt{b^2 - 4ac}}{2a}[/tex]
Where:
[tex]a = -0.5; b= 36; c = 132[/tex]
So, we have:
[tex]x = \frac{-36 \± \sqrt{36^2 - 4*-0.5*132}}{2*-0.5}[/tex]
[tex]x = \frac{-36 \± \sqrt{1560}}{-1}[/tex]
[tex]x = \frac{-36 \± 39.50}{-1}[/tex]
Split
[tex]x = \frac{-36 + 39.50}{-1}; x = \frac{-36 - 39.50}{-1}[/tex]
[tex]x = \frac{3.50}{-1}; x = \frac{-75.50}{-1}[/tex]
[tex]x = 75.50[/tex] and [tex]x = -3.50[/tex]
The amount can't be negative.
So:
[tex]x = 75.50[/tex]
Hence, the least amount is $75.50
