Test Index

Units and Measurements

Section: Physics

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Question : 1 of 13

Marks: +1, -0

If the error in the measurement of the surface area of a sphere is

1.2\%

, then the error in the determination of the volume of the sphere is

[AP EAPCET 24 May 2025 S1]

$\;2.4\%$
$\;1.8\%$
$\;1.2\%$
$\;0.6\%$

Solution:

Error in surface area

S=1.2\%

We need to find the percentage error in volume

V

of a sphere.

Relations:

For a sphere of radius

r

:

\;S=4\pi r^2

\;V=\;\frac{4}{3}\pi r^3

Step 1: Express relative errors

For small errors,

\;\;\frac{dS}{S}=2\;\frac{dr}{r}

\;\;\frac{dV}{V}=3\;\frac{dr}{r}

Step 2: Connect the two

From the first equation:

\;\frac{dr}{r}=\;\frac{1}{2}\;\frac{dS}{S}

Substitute into the second:

\;\frac{dV}{V}=3\times\;\frac{1}{2}\;\frac{dS}{S}=\;\frac{3}{2}\;\frac{dS}{S}

Step 3: Substitute given error

Error in

V=\;\frac{3}{2}\times1.2\%=1.8\%

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