If the radius of a sphere is measured as 9 cm with an error of 0.03 m, find the approximate error in calculating its surface area.

Given the radius of a sphere is measured as 9 cm with an error of 0.03 m = 3 cm.

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

Hence, we have x = 9 and Δx = 3

The surface area of a sphere of radius x is given by

S = 4πx²

On differentiating S with respect to x, we get

We know

When x = 9, we have.

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 = 3

⇒ ΔS = (72π)(3)

∴ ΔS = 216π

Thus, the approximate error in calculating the surface area of the sphere is 216π cm².