# Percentage error in diameter given percentage error in surface area.

I am given a cylinder with a height of 10m.

I am tasked with finding the maximum percentage error in measuring the exterior diameter of the tank, given that the maximum error in surface area is 8%.

I have gotten to $\frac{(4{\pi}r+20\pi)dr}{(2{\pi}r^2+20{\pi}r)}=.08$ but I am not sure how to continue.

Any help would be greatly appreciated. Thank you.

• Perhaps you mean to estimate the diameter by measuring the surface area? – copper.hat Oct 4 '16 at 0:40

The surface area is given by $S(d) = \pi( dh + {1 \over 2} d^2)$. Given a surface area $A\ge 0$, there is a unique $d$ such that $S(d) = A$ and this is given by $\delta(A) = \sqrt{2 {A \over \pi} +h^2}-h$.
Hence if $A$ is the correct area and $A(1+ \epsilon)$ is the measured area, you want to compute $\Delta=\max_{|\epsilon| \le 0.08} | {\delta(A(1+\epsilon)) \over \delta(A)} -1|$. Since $\delta$ is an increasing function and it's derivative $\delta'$ is decreasing, we see that $\Delta = 1-{\delta(A(1-0.08)) \over \delta(A)}$.