Does differentiating volume give surface area?

Question

Hi there,

In this question, you have to find the rate at which the surface area of the water is increasing.

I understand up to the step where you find the volume of the truncated pyramid.

But how come the surface area formula is the same for the differentiated formula for the pyramid? Are they supposed to be the same?

Thanks

Answer 1

Your question relates to the fundamental theorem of calculus, which you probably haven't learned yet.

As you add water to the vessel you can think of it filling out that red layer on the surface. The thickness of the surface is small. So, the change in volume is proportional to the area of the surface.

I won't say that the change in volume is always the surface area, because it does depend a little bit on how you set it up and which surfaces you are comparing to. i.e. The lateral surfaces are not be considered here.

But frequently the change in volume is the surface area.