The revenue R from sale of x units of a commodity is given by R = 20 x - 0.5 `x^2` . Percentage rate of change of R when x = 10 is (i) `15%` (ii) `6(2)/3%` (iii) `1/15%` (iv) `20%`

Options

A. `15%`
B. `6(2)/3%`
C. `1/15%`
D. `20%`

Answer

To find the percentage rate of change of revenue \( R \) when \( x = 10 \), we will follow these steps:

### Step 1: Write the Revenue Function
The revenue \( R \) from the sale of \( x \) units of a commodity is given by:
\[ R = 20x - 0.5x^2 \]

### Step 2: Differentiate the Revenue Function
To find the rate of change of revenue with respect to \( x \), we need to differentiate \( R \) with respect to \( x \):
\[ \frac{dR}{dx} = \frac{d}{dx}(20x - 0.5x^2) \]
Using the power rule of differentiation:
\[ \frac{dR}{dx} = 20 - 0.5 \cdot 2x = 20 - x \]

### Step 3: Evaluate the Derivative at \( x = 10 \)
Now, we will substitute \( x = 10 \) into the derivative:
\[ \frac{dR}{dx} \bigg|_{x=10} = 20 - 10 = 10 \]

### Step 4: Calculate the Revenue at \( x = 10 \)
Next, we need to find the revenue \( R \) when \( x = 10 \):
\[ R \bigg|_{x=10} = 20(10) - 0.5(10^2) = 200 - 50 = 150 \]

### Step 5: Find the Relative Rate of Change
The relative rate of change of \( R \) is given by:
\[ \text{Relative Rate of Change} = \frac{\frac{dR}{dx}}{R} = \frac{10}{150} \]

### Step 6: Convert to Percentage
To convert the relative rate of change to a percentage, we multiply by 100:
\[ \text{Percentage Rate of Change} = \left(\frac{10}{150}\right) \times 100 = \frac{1000}{150} = \frac{20}{3} \approx 6.67\% \]
This can also be expressed as \( 6 \frac{2}{3}\% \).

### Conclusion
The percentage rate of change of \( R \) when \( x = 10 \) is:
\[ \boxed{6 \frac{2}{3}\%} \]