The circumference of a sphere was measured to be 89cm with a possible error of 0.5cm. Use linear approximation to estimate the maximum error in the calculated surface area.

Question

I understand that linear approximation is f(a)+f'(a)(x-a). I know that the surface area is 2pir^2. I can't seem to puzzle together how everything fits. Can somebody explain HOW to solve this problem using linear approximation?? step-by-step, because I can't seem to find anybody who didn't just spit out an equation and an answer and I'm really confused.

Answer 1

Surface area of a sphere, A = 4πr²

If C=2πr then A=C²/π

dA/dC =2C/π

dA = (2/π)CdC

Using approximation, where C= 89 and ΔC = 0.5

Maximum area = ΔA= (2/π)CΔC = (2/π)(89)(0.5) ≈ 28.33 cm²

Relative area = ΔA/A =28.33/2521.33 = 1.12%