To find the correct statement regarding the difference between the maximum profits earned by the sandwich shop and the smoothie stand, we first need to determine the maximum profits for each business.
### Step-by-Step Solution:
1. Profit Functions:
- For the sandwich shop:
[tex]\[
f(x) = -500x^2 + 5000x + 7500
\][/tex]
- For the smoothie stand:
[tex]\[
g(x) = -300x^2 + 3600x + 6000
\][/tex]
2. Finding the Vertex (Maximum Profit):
The vertex form of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is found at:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
- For the sandwich shop profit function [tex]\( f(x) \)[/tex]:
[tex]\[
a = -500, \quad b = 5000
\][/tex]
[tex]\[
x = -\frac{5000}{2 \times -500} = 5
\][/tex]
Substitute [tex]\( x = 5 \)[/tex] back into [tex]\( f(x) \)[/tex]:
[tex]\[
f(5) = -500(5)^2 + 5000(5) + 7500 = -500(25) + 25000 + 7500 = -12500 + 25000 + 7500 = 20000
\][/tex]
So, the maximum profit for the sandwich shop is \[tex]$20,000.
- For the smoothie stand profit function \( g(x) \):
\[
a = -300, \quad b = 3600
\]
\[
x = -\frac{3600}{2 \times -300} = 6
\]
Substitute \( x = 6 \) back into \( g(x) \):
\[
g(6) = -300(6)^2 + 3600(6) + 6000 = -300(36) + 21600 + 6000 = -10800 + 21600 + 6000 = 16800
\]
So, the maximum profit for the smoothie stand is \$[/tex]16,800.
3. Calculating the Difference in Maximum Profits:
[tex]\[
\text{Difference} = 20000 - 16800 = 3200
\][/tex]
Therefore, the correct statement is:
D. The difference between the maximum profit earned by the sandwich shop and the smoothie stand is \$3,200.
