If there is an error of 0.01 cm in the diameter of a sphere, then percentage error in surface area when the radius =5 cm, is

Options

A. `0.005%`
B. `0.05%`
C. `0.1%`
D. `0.2%`

Answer

To solve the problem of finding the percentage error in the surface area of a sphere given an error in the diameter, we can follow these steps:

### Step 1: Understand the given values
We are given:
- Error in diameter (ΔD) = 0.01 cm
- Radius (R) = 5 cm

### Step 2: Calculate the diameter
The diameter (D) of the sphere can be calculated using the formula:
\[ D = 2R \]
Substituting the value of R:
\[ D = 2 \times 5 = 10 \text{ cm} \]

### Step 3: Write the formula for surface area
The surface area (S) of a sphere is given by the formula:
\[ S = 4\pi R^2 \]

### Step 4: Differentiate the surface area
To find the differential of the surface area, we differentiate S with respect to R:
\[ dS = \frac{dS}{dR} \cdot dR = 8\pi R \cdot dR \]

### Step 5: Relate the changes in diameter and radius
Since the diameter is related to the radius by the equation \( D = 2R \), we can differentiate this:
\[ dD = 2 \cdot dR \]
This implies:
\[ dR = \frac{dD}{2} \]

### Step 6: Substitute the error in diameter
Now, substituting the error in diameter (ΔD = 0.01 cm):
\[ dR = \frac{0.01}{2} = 0.005 \text{ cm} \]

### Step 7: Calculate the percentage error in surface area
Using the formula for the percentage error in surface area:
\[ \frac{dS}{S} = 2 \cdot \frac{dR}{R} \]

Substituting the known values:
- R = 5 cm
- \( dR = 0.005 \text{ cm} \)

Calculating \( \frac{dR}{R} \):
\[ \frac{dR}{R} = \frac{0.005}{5} = 0.001 \]

Now substituting this back into the percentage error formula:
\[ \frac{dS}{S} = 2 \cdot 0.001 = 0.002 \]

### Step 8: Convert to percentage
To find the percentage error:
\[ \text{Percentage Error} = \frac{dS}{S} \times 100 = 0.002 \times 100 = 0.2\% \]

### Conclusion
The percentage error in the surface area of the sphere when the radius is 5 cm and there is an error of 0.01 cm in the diameter is **0.2%**.