Hello, xje4bv!

If these "differential approximations" are confusing to you, don't give up.

It took me the looongest time to understand them.

A globe with a diameter of 30 cm is to be painted

with a uniform layer of paint which is 4 mm thick.

Use the method of differentiation to find the approximate volume of paint needed.

We have a sphere with radius 15 cm.

It has a volume according to the formula: .

By adding a layer of paint, the radius is increased by

The question is: by how much is the volume increased?

We could do the actual arithmetic,

. . but we canapproximatethe answer with differentials.

We have: .

Take differentials: .

If we understand what each part means, we're all set!

. .

We know: .

Therefore: .

The volume of the paint needed isapproximately1131 cm³