A coat of paint of thickness $0.07$ cm is to be applied uniformly to the faces of a cube of edge $37$ cm. Use differentials...

Question

A coat of paint of thickness $0.07$ cm is to be applied uniformly to the faces of a cube of edge $37$ cm. Use differentials to find the approximate amount of paint required for the job, correct to the nearest cubic centimeter.

I got this question on my math homework and I got to

A'=12x 12x*dx=6(x+^x)^2-6(x)2

What do I do? My final answer was $498$ but the correct one is $575$. The correct equation was 3(x)^2*^x how do I get to that?

Answer 1

Let$ \ s \ $be the edge of the cube.

Thus:

$$ \ s=37\ cm \ $$

After the paint is applied, the edge of the cube will change from$\ 37\ cm $ to $ 37.14\ cm $:

$$ \ ds=0.14\ cm \ $$

Recall the formula for volume of a cube:

$$ \ V=s^{3} \ $$

Differentiating, we get:

$$ \ dV=3s^{2}ds \ $$

"Plugging in":

$$ \ dV=3(37\ cm)^{2}(0.14\ cm) \ $$

Which gives us:

$$ \ dV=574.98\ cm^{3} \ $$

Therefore, the volume of the cube changes by approximately$ \ 575\ cm^{3} \ $