Let's analyze the given functions step-by-step in order to determine the two correct statements about the maximum profit and the corresponding selling price.
### Step 1: Define the Functions
First, we have the revenue and cost functions for the headphones:
[tex]\[
R(x) = -210x^2 + 6970x
\][/tex]
[tex]\[
C(x) = -170x + 39690
\][/tex]
### Step 2: Profit Function
The profit function [tex]\( P(x) \)[/tex] is given by the revenue function minus the cost function:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Substituting the given functions:
[tex]\[
P(x) = (-210x^2 + 6970x) - (-170x + 39690)
\][/tex]
[tex]\[
P(x) = -210x^2 + 6970x + 170x - 39690
\][/tex]
[tex]\[
P(x) = -210x^2 + 7140x - 39690
\][/tex]
### Step 3: Find the Maximum Profit
The maximum profit for a parabolic function [tex]\( P(x) = ax^2 + bx + c \)[/tex] occurs at its vertex. The [tex]\( x \)[/tex]-coordinate of the vertex of a parabola is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Here, [tex]\( a = -210 \)[/tex] and [tex]\( b = 7140 \)[/tex]. Thus:
[tex]\[
x = -\frac{7140}{2 \cdot -210} = \frac{7140}{420} = 17
\][/tex]
So, the maximum profit occurs when [tex]\( x = 17 \)[/tex].
Now, we can check the value of the maximum profit by substituting [tex]\( x = 17 \)[/tex] back into the profit function [tex]\( P(x) \)[/tex]:
[tex]\[
P(17) = -210(17)^2 + 7140(17) - 39690
\][/tex]
[tex]\[
P(17) = -210 \cdot 289 + 7140 \cdot 17 - 39690
\][/tex]
[tex]\[
P(17) = -60690 + 121380 - 39690
\][/tex]
[tex]\[
P(17) = 21000
\][/tex]
So, the maximum profit is [tex]$\$[/tex]21,000[tex]$.
### Step 4: Selling Price at Maximum Profit
To find the selling price when the maximum profit occurs, we know the revenue function \( R(x) \), and we understand that the function gives total revenue for \( x \) units sold. Therefore, the average selling price \( S \) at \( x = 17 \) units can be calculated as:
\[
S = \frac{R(17) - R(0)}{17}
\]
Calculating \( R(17) \):
\[
R(17) = -210(17)^2 + 6970(17)
\]
\[
R(17) = -210 \cdot 289 + 6970 \cdot 17
\]
\[
R(17) = -60690 + 118490
\]
\[
R(17) = 57800
\]
Calculating \( S \):
\[
S = \frac{57800 - 0}{17}
\]
\[
S = \frac{57800}{17}
\]
\[
S = 3400
\]
Thus, the selling price at the maximum profit of \$[/tex]21,000 is \[tex]$3,400.
### Conclusion
Based on this analysis, the two correct statements are:
- The maximum profit is $[/tex]\[tex]$21,000$[/tex].
- A selling price of \[tex]$3400 results in the maximum profit.
Therefore, the correct statements from the provided options would be:
- The maximum profit is \$[/tex]21,000.
- A selling price of \$3400 results in the maximum profit.
