Let's analyze the given functions and the solution step by step for clarity.
The revenue function [tex]\( R(x) \)[/tex] and the cost function [tex]\( C(x) \)[/tex] are given as:
[tex]\[
R(x) = -210x^2 + 8970\pi
\][/tex]
[tex]\[
C(x) = -170x + 39690
\][/tex]
To find the maximum profit, we need to determine the profit function [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
First, we find the profit function:
[tex]\[
P(x) = (-210x^2 + 8970\pi) - (-170x + 39690)
\][/tex]
Simplifying the profit function:
[tex]\[
P(x) = -210x^2 + 8970\pi + 170x - 39690
\][/tex]
To find the critical point where the profit is maximized, we take the first derivative of [tex]\( P(x) \)[/tex] and set it to zero:
[tex]\[
P'(x) = d(-210x^2 + 8970\pi + 170x - 39690) / dx
\][/tex]
[tex]\[
P'(x) = -420x + 170
\][/tex]
Setting the first derivative equal to zero to find the critical point:
[tex]\[
0 = -420x + 170
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{170}{420} \approx 0.4048
\][/tex]
Next, we plug this critical value back into the original revenue and cost functions to find the corresponding revenue, cost, and ultimately the profit at this critical point.
Revenue at [tex]\( x = 0.4048 \)[/tex]:
[tex]\[
R = -210 \times (0.4048)^2 + 8970\pi \approx 28145.68
\][/tex]
Cost at [tex]\( x = 0.4048 \)[/tex]:
[tex]\[
C = -170 \times 0.4048 + 39690 \approx 39621.19
\][/tex]
Maximum profit:
[tex]\[
\text{Profit} = R - C = 28145.68 - 39621.19 = -11475.51
\][/tex]
The selling price at this [tex]\( x \)[/tex] is:
[tex]\[
\text{Selling price} = -170 \times 0.4048 + 39690 \approx 39621.19
\][/tex]
Now let's analyze the statements:
- "The maximum profit is \[tex]$57,834." - This statement is false because the maximum profit calculated is approximately -\$[/tex]11,475.51.
- "The maximum profit is \[tex]$21,000." - This statement is false as well for the same reason above.
- "A selling price of \$[/tex]27 results in the maximum profit." - This statement is false because the selling price for maximum profit is approximately \[tex]$39,621.19.
- "A selling price of \$[/tex]17 results in the maximum profit." - This statement is also false for the same reason above.
- "A selling price of \$16.60 results in the maximum profit." - This statement is false for the same reason above.
None of the statements mentioned above are true based on the given functions and the solution.
