graysonmarie12
Answered

Select the correct answer.

Mr. Olaffsen opened a sandwich shop and a smoothie stand in his neighborhood. The following table and equation show function [tex]f[/tex], representing Mr. Olaffsen's profit, in dollars, [tex]x[/tex] months since opening the sandwich shop.

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
[tex]x[/tex] & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
[tex]f(x)[/tex] & 12,000 & 15,500 & 18,000 & 19,500 & 20,000 & 19,500 & 18,000 \\
\hline
\end{tabular}

[tex]
f(x)=-500 x^2+5,000 x+7,500
[/tex]

The following table and equation show function [tex]g[/tex], representing Mr. Olaffsen's profit, in dollars, [tex]x[/tex] months since opening the smoothie stand.

\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
[tex]x[/tex] & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
[tex]g(x)[/tex] & 9,300 & 12,000 & 14,100 & 15,600 & 16,500 & 16,800 & 16,500 \\
\hline
\end{tabular}

[tex]
g(x)=-300 x^2+3,600 x+6,000
[/tex]

Select the true statement.

A. The difference between the maximum profit earned by the sandwich shop and the smoothie stand is [tex]\$3,000[/tex].

B. The difference between the maximum profit earned by the sandwich shop and the smoothie stand is [tex]\$2,700[/tex].

C. The difference between the maximum profit earned by the sandwich shop and the smoothie stand is [tex]\$3,500[/tex].

D. The difference between the maximum profit earned by the sandwich shop and the smoothie stand is [tex]\$3,200[/tex].



Answer :

GinnyAnswer
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.

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