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

Answer:

δV/δr,dh   = 14.06 cm³

Step-by-step 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