# Math Help - Linear approximation

1. ## Linear approximation

I need help with this problem, i dont know how to set it up.

Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.02000 cm thick to a hemispherical dome with a diameter of 45.00 meters.

2. Originally Posted by viet
I need help with this problem, i dont know how to set it up.

Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint 0.02000 cm thick to a hemispherical dome with a diameter of 45.00 meters.
We have that volume of the hemisphere is a function of the radius, that is,
$V(r)=\frac{2}{3}\pi r^3$
Since $.02$ is a small number we have that the change in the derivative is approximately the change in the actual function, that is,
$dV\approx \Delta V$
Since,
$\frac{dV}{dr}=2\pi r^2$
The change in derivative is,
$dV=2\pi r^2 dr$
Where $dr=\Delta r$ the change in the radius.
Which is, $+.02$
Thus,
$dV=2\pi (45)^2 (.02)$
Thus,
$dV=2\pi (2025)(.02)$
Thus,
$dV=81 \pi$
That means the change in volume is approximately increased by $81 \pi\mbox{ cm}$