Question: Question 3 of 12, Step 1 of 1 The revenue function is given by \( R(x) = x \cdot p(x) \)…

Question 3 of 12, Step 1 of 1

The revenue function is given by \( R(x) = x \cdot p(x) \) dollars where \( x \) is the number of units sold and \( p(x) \) is the unit price. If \( p(x) = 42(4)^{\frac{x}{4}} \), find the revenue if 12 units are sold. Round to two decimal places.

Answer

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Solution

The revenue function is given by \( R(x) = x \cdot p(x) \), where \( x \) is the number of units sold and \( p(x) \) is the unit price. We need to find the revenue if 12 units are sold and \( p(x) = 42(4)^{\frac{x}{4}} \). First, substitute \( x = 12 \) into the equation for \( R(x) \). The expression for \( p(x) \) is: \[ p(x) = 42 \cdot 4^{\frac{x}{4}} \] Substitute \( x = 12 \) into \( p(x) \): \[ p(12) = 42 \cdot 4^{\frac{12}{4}} \] Calculate \( \frac{12}{4} \): \[ \frac{12}{4} = 3 \] Now calculate \( 4^3 \): \[ 4^3 = 64 \] Substitute back to find \( p(12) \): \[ p(12) = 42 \cdot 64 \] Calculate the multiplication: \[ p(12) = 2688 \] Now calculate \( R(12) \): \[ R(12) = 12 \cdot 2688 \] Calculate the final revenue: \[ R(12) = 32256 \] Thus, the revenue when 12 units are sold is $32256.00.