Question

The volume V of a right‑circular cylinder is computed using the values d = 3.1 m for the diameter and h = 6.6 m for the height. Use the Linear Approximation to estimate the maximum error R in V if each of these values has a possible error of at most 2 % . Recall that V = π r 2 h .

Answer 1

Answer:

6%

Step-by-step explanation:

A function f(x) can by approximated at and around a point, say x = a by,

f(x) = f(a) + f'(a)(x-a)

where,  $f'(x)=\frac{df}{dx}$

Here, volume V is a function of radius r and height h. In the 2 dimensional case, we have to take the partial derivatives.

Given,

$V=\pi r^2h$

or, $dV=\frac{\partial V }{\partial r}dr+ \frac{\partial V }{\partial h}dh$

at r = 1.55 m and h = 6.6 m

$dr=0.02\times1.55m=0.031m$

$dh=0.02\times6.6m=0.132m$

$\frac{\partial V }{\partial r}=2\pi r h=2\pi \times 1.55\times 6.6=64.277$

$\frac{\partial V }{\partial h}=\pi r^2=7.5477$

Therefore, $dV=(64.277\times.031)+(7.5477\times0.132)=2.9889m$

Also, $V=\pi r^2h=49.8147m$

Hence, maximum error is given by,

$R=\frac{2.9889}{49.8147} \times 100$

= 6%