# Linear approximation

• Oct 27th 2006, 10:03 AM
viet
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.
• Oct 27th 2006, 10:20 AM
ThePerfectHacker
Quote:

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,
$\displaystyle V(r)=\frac{2}{3}\pi r^3$
Since $\displaystyle .02$ is a small number we have that the change in the derivative is approximately the change in the actual function, that is,
$\displaystyle dV\approx \Delta V$
Since,
$\displaystyle \frac{dV}{dr}=2\pi r^2$
The change in derivative is,
$\displaystyle dV=2\pi r^2 dr$
Where $\displaystyle dr=\Delta r$ the change in the radius.
Which is, $\displaystyle +.02$
Thus,
$\displaystyle dV=2\pi (45)^2 (.02)$
Thus,
$\displaystyle dV=2\pi (2025)(.02)$
Thus,
$\displaystyle dV=81 \pi$
That means the change in volume is approximately increased by $\displaystyle 81 \pi\mbox{ cm}$