If the demand function is `x= (24-2p)/(3)`where x is the number of units produced and p is the price per unit, then the revenue function R(x) is

Options

A. `R(x)=12x-(3)/(2)x^(2)`
B. `R(x)=8x-(2)/(3)x^(2)`
C. `R(x)=(2)/(3)x^(2)-12x`
D. `R(x)=24x+(3)/(2)x^(2)`

Answer

To find the revenue function \( R(x) \) given the demand function \( x = \frac{24 - 2p}{3} \), we will follow these steps:

### Step 1: Express the price \( p \) in terms of \( x \)
We start with the demand function:
\[ x = \frac{24 - 2p}{3} \]
To express \( p \) in terms of \( x \), we can rearrange this equation:
\[ 3x = 24 - 2p \]
Now, isolate \( p \):
\[ 2p = 24 - 3x \]
\[ p = \frac{24 - 3x}{2} \]

### Step 2: Write the revenue function \( R(x) \)
The revenue function \( R(x) \) is given by the product of the price per unit \( p \) and the number of units produced \( x \):
\[ R(x) = p \cdot x \]
Substituting the expression for \( p \) from Step 1:
\[ R(x) = \left(\frac{24 - 3x}{2}\right) \cdot x \]

### Step 3: Simplify the revenue function
Now, we will simplify the expression:
\[ R(x) = \frac{(24 - 3x) \cdot x}{2} \]
Distributing \( x \):
\[ R(x) = \frac{24x - 3x^2}{2} \]
Now, we can separate the terms:
\[ R(x) = \frac{24x}{2} - \frac{3x^2}{2} \]
This simplifies to:
\[ R(x) = 12x - \frac{3}{2}x^2 \]

### Final Revenue Function
Thus, the revenue function is:
\[ R(x) = 12x - \frac{3}{2}x^2 \]