Question

Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick.

Answer 1

δV/δr,dh = 14.06 cm³

Explanation:

The volume of the can is:

Vc = π*r²*h where r is the radius of the base and h is the heigth

If we take partial derivatives for that equation we get:

δV/δr = 2*π*r*h or δV = 2*π*r*h*dr

δV/δh = 2*π*r² or δV = 2*π*r²*dh

Now the can varies its height at the top and the bottom then:

dh = 0.04cm*2 = 0.08 cm

And

dr = 0.04

δV/δr,dh = 2*π*r*h*dr + 2*π*r²*dh

By substitution:

δV/δr,dh = 2*4*6*π*(0.04) + 32*π*(0.08)

δV/δr,dh = 1.92*π + 2.56*π

δV/δr,dh = 14.06 cm³