The percentage error in the calculation of the volume of a sphere is 4.8%. What is the percentage error in the measurement of radius of the sphere.

Explanation

To solve the problem of finding the percentage error in the measurement of the radius of a sphere given the percentage error in the calculation of its volume, we can follow these steps:

### Step 1: Understand the relationship between volume and radius
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Determine the formula for percentage error in volume
The percentage error in volume can be expressed in terms of the percentage error in radius. The formula for percentage error in volume is:
\[ \text{Percentage Error in } V = \frac{\Delta V}{V} \times 100 \]
where \( \Delta V \) is the absolute error in volume.

### Step 3: Relate the errors
For a function of the form \( V = k r^n \) (where \( k \) is a constant and \( n \) is the power of \( r \)), the percentage error in \( V \) can be related to the percentage error in \( r \) as follows:
\[ \text{Percentage Error in } V = n \times \text{Percentage Error in } r \]
In the case of the volume of a sphere, \( n = 3 \).

### Step 4: Set up the equation
Given that the percentage error in volume is \( 4.8\% \), we can set up the equation:
\[ 4.8 = 3 \times \text{Percentage Error in } r \]

### Step 5: Solve for the percentage error in radius
To find the percentage error in radius, we rearrange the equation:
\[ \text{Percentage Error in } r = \frac{4.8}{3} \]
Calculating this gives:
\[ \text{Percentage Error in } r = 1.6\% \]

### Conclusion
The percentage error in the measurement of the radius of the sphere is \( 1.6\% \).

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