# Recall that the volume of a sphere of radius r is V(r) = \frac{4}{3} \pi r^3 . Find L, the...

## Question:

Recall that the volume of a sphere of radius {eq}r {/eq} is {eq}V(r) = \frac{4}{3} \pi r^3 {/eq} .

Find {eq}L {/eq}, the linearisation of {eq}V(r) {/eq} at {eq}r = 50 {/eq}.

## Linearization Of A Function:

Given a function {eq}\displaystyle f(x) {/eq} which is differentiable at {eq}\displaystyle x=a {/eq} can be linearized as follows

$$\displaystyle L(x)\approx f(a)+f'(a)(x-a)$$

Given the formula for volume of a sphere we have

{eq}\displaystyle V=\frac{4}{3}\pi r^3\\ \displaystyle \Rightarrow \frac{dV}{dr}=4\pi r^2 {/eq}

Here we can see that the volume function is differentiable even at {eq}\displaystyle r=50 {/eq}

So the linearization of the volume function at {eq}\displaystyle r=50 {/eq} can be written as

{eq}\displaystyle \begin{align} L(r)&\approx V(50)+\frac{dV}{dr}\big|_{r=50}(r-50)\\ &=\left[ \frac{4}{3}\pi (50)^3\right]+\left[ 4\pi r^2\right]_{r=50}(r-50)\\ &=\frac{4}{3}\pi 125000+\left[ 4\pi(50)^2\right](r-50)\\ &=\frac{500000\pi}{3}+10000\pi(r-50)\\ &=10000\pi\left[(r-50)+\frac{50}{3} \right]\\ &=10000\pi\left[ r-\frac{100}{3}\right] \end{align} {/eq} 