Alright, let's solve the problem step-by-step.
1. Determine the unit price [tex]\( p(x) \)[/tex]:
Given the unit price function:
[tex]\[
p(x) = 27 \cdot 4^{\frac{\pi}{5}}
\][/tex]
We need to first evaluate the expression [tex]\( 4^{\frac{\pi}{5}} \)[/tex].
2. Evaluate [tex]\( 4^{\frac{\pi}{5}} \)[/tex]:
We know that [tex]\( \pi \approx 3.14159 \)[/tex]. Let's substitute this value into the exponent:
[tex]\[
4^{\frac{3.14159}{5}}
\][/tex]
3. Calculate [tex]\( 4^{\frac{3.14159}{5}} \)[/tex]:
Using the provided approximation, we find:
[tex]\[
4^{\frac{3.14159}{5}} \approx 2.3893813336
\][/tex]
4. Find [tex]\( p(x) \)[/tex]:
Substitute this back into the unit price function:
[tex]\[
p(x) = 27 \cdot 2.3893813336 \approx 64.51324595716412
\][/tex]
This gives us the unit price to be approximately [tex]\( p(x) \approx 64.51 \)[/tex] dollars (to two decimal places).
5. Calculate the total revenue:
Now, we need to calculate the revenue for 10 units sold. The revenue function is:
[tex]\[
R(x) = x \cdot p(x)
\][/tex]
Given [tex]\( x = 10 \)[/tex]:
[tex]\[
R(10) = 10 \cdot 64.51324595716412 \approx 645.1324595716412
\][/tex]
6. Round the revenue to two decimal places:
The calculated revenue is approximately 645.1324595716412 dollars. Rounding this to two decimal places, we get:
[tex]\[
R(10) \approx 645.13 \text{ dollars}
\][/tex]
Thus, the revenue if 10 units are sold, rounded to two decimal places, is [tex]\( 645.13 \)[/tex] dollars.
