To find the revenue when 12 units are sold, we need to follow these steps:
1. Determine the unit price, [tex]\( p(x) \)[/tex], given by the function [tex]\( p(x) = 42 \times 4^{\frac{-2}{4}} \)[/tex].
Let's simplify [tex]\( 4^{\frac{-2}{4}} \)[/tex]:
[tex]\[
4^{\frac{-2}{4}} = 4^{-\frac{1}{2}} = \frac{1}{4^{\frac{1}{2}}} = \frac{1}{\sqrt{4}} = \frac{1}{2}
\][/tex]
Therefore, substituting back into the function, we get:
[tex]\[
p(x) = 42 \times \frac{1}{2} = 21
\][/tex]
2. Next, we calculate the revenue, [tex]\( R(x) \)[/tex], where [tex]\( x = 12 \)[/tex] units. The revenue function is given by:
[tex]\[
R(x) = x \times p(x)
\][/tex]
Substituting the values:
[tex]\[
R(12) = 12 \times 21 = 252
\][/tex]
3. Finally, since we need to round the revenue to two decimal places, we observe that 252 is already an integer, so:
[tex]\[
R(12) = 252.00
\][/tex]
So, the unit price, the total revenue, and the rounded revenue when 12 units are sold are:
- Unit price, [tex]\( p(x) \)[/tex]: \[tex]$21.00
- Total revenue: \$[/tex]252.00
- Rounded revenue (to two decimal places): \$252.00
