# Math Help - Differentials to calculate Maximum Error

1. ## Differentials to calculate Maximum Error

The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm. Use differentials to estimate the maximum error in the calculated volume.

2. Hello, drahcirnaw!

I had to snake my way through this one.
Hope I got it right . . .

The circumference of a sphere was measured to be 90 cm with a possible error of 0.5 cm.
Use differentials to estimate the maximum error in the calculated volume.

We have: . $C \:=\:2\pi r$

We are given: . $C = 90 \quad\Rightarrow\quad 90 \:=\:2\pi r \quad\Rightarrow\quad r \:=\:\frac{45}{\pi}$ .[1]

We have: . $dC \:=\:2\pi dr$

We are given: . $dC \,=\,\pm0.5 \quad\Rightarrow\quad \pm 0.5 \:=\:2\pi dr \quad\Rightarrow\quad dr \:=\:\frac{\pm0.5}{2\pi}$ .[2]

The volume of a sphere is: . $V \:=\:\frac{4}{3}\pi r^3$

. . Then: . $dV \;=\;4\pi r^2dr$

Substitute [1] and [2]: . $dV \;=\;4\pi\left(\frac{45}{\pi}\right)^2\left(\frac{ \pm0.5}{2\pi}\right) \;=\;\pm\frac{2025}{\pi^2}$

Therefore, the maximum error in volume is about: . $\pm205.2\text{ cm}^3$

3. Thank you so much Soroban. Makes much more sense now.