Question

If the error in the measurement of the volume of a sphere is 6%, then find the error in the measurement of its surface area.
A. 2%
B. 3%
C. 4%
D. 7.5%

Answer 1

Hint: Error is defined as the difference between two measurements. The error in measurement is nothing but a mathematical way to show the uncertainty in the measurement and it is the difference between the result of the measurement and the true value.





Formula used:
To find the volume of the sphere the formula is,
\[V = \dfrac{4}{3}\pi {r^3}\]
Where,
r is radius of the sphere

To find the surface area of sphere the formula is,
\[S = 4\pi {r^2}\]
Where,
r is radius of the sphere




Complete answer:
Consider a sphere in which if the volume of a sphere has an error of 6%, then we need to find the error in the measurement of its surface area.
In order to calculate the error, we can write it as,
\[\left( {\dfrac{{\Delta V}}{V} \times 100} \right) = \left( {3\dfrac{{\Delta r}}{r} \times 100} \right)\]
Substitute the value of error in the volume in the above equation, we get,
\[6\% = \left( {3\dfrac{{\Delta r}}{r} \times 100} \right)\]
Since \[\dfrac{4}{3}\pi \]is a constant, we eliminate it while calculating the error because it only depends on the radius of the sphere.
\[2\% = \left( {\dfrac{{\Delta r}}{r} \times 100} \right)\]
The percentage error in the measurement of the radius is 2%.
In order to find the error in the measurement of the surface area, we have,
\[\left( {\dfrac{{\Delta S}}{S} \times 100} \right) = \left( {2\dfrac{{\Delta r}}{r} \times 100} \right)\]
The percentage error in the measurement of the surface area is
=2% X error in radius
2% X 2% = 4%
Therefore, the error in the measurement of its surface area is 4%

Hence, Option C is the correct answer


Note: Here, in this problem, it is important to know the formula for the volume of the sphere, the surface area of the sphere, and how to calculate the error in the percentage as shown in this solution.