Q. 53.7( 3 Votes )

If there is an error of 0.1% in the measurement of the radius of a sphere, find approximately the percentage error in the calculation of the volume of the sphere.

Answer :

Given the error in the measurement of the radius of a sphere is 0.1%.


Let x be the radius of the sphere and Δx be the error in the value of x.


Hence, we have


Δx = 0.001x


The volume of a sphere of radius x is given by



On differentiating V with respect to x, we get




We know




Recall that if y = f(x) and Δx is a small increment in x, then the corresponding increment in y, Δy = f(x + Δx) – f(x), is approximately given as



Here, and Δx = 0.001x


ΔV = (4πx2)(0.001x)


ΔV = 0.004πx3


The percentage error is,




Error = 0.003 × 100%


Error = 0.3%


Thus, the error in calculating the volume of the sphere is 0.3%.


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