# Using differentials to find the maximum error that a diameter can have so that the area error is within $1\%$

I am given the following problem:

The area of a circle was computed using the measurement of its diameter. Use differentials to find the maximum error that a diameter can have so that the area error is within $1\%$

What I have so far is

$$A = \pi r^2 = \frac{\pi d^2}{4}\\ \\ \frac{dA}{dd} = \frac{\pi}{4} 2d\\ dA = \frac{\pi}{2} d \ dd\\ dd = \frac{0.1 d}{2} = 0.05 \cdot d = 5\% \cdot d$$

So the answer would be $5\%$.

Is that correct? Am I missing something?

Thank you.

• You lost a $0$, $1/100=0.01$ not $0.1$ – G Cab May 28 '16 at 23:36

## 1 Answer

The area, given the diameter, is $$A(d) = \frac{\pi d^2}{4}.$$ Therefore $$A(d+\Delta d) = \frac{\pi}{4}(d^2+2d\Delta d + \Delta d^2).$$ If the (I assume, relative) area error is less than $1\%$, $$0.01 \leq \left|\frac{A(d+\Delta d)-A(d)}{A(d)}\right| = \left|\frac{2d\Delta d + \Delta d^2}{d^2}\right|$$

The diameter error is $$E = \left|\frac{d + \Delta d - d}{d}\right| = \left|\frac{\Delta d}{d}\right|$$ so that $$0.01 \leq 2 E + E^2.$$

Now can you find which values of $E$ satisfy the inequality?