The edges of a cube are 24 inches and are increasing at the rate of .02 in per minute. At what rate is a) the volume increasing? B) the are increasing?

Question

Thank you for the help!

Answer 1

Volume (V) equals the side of the cube (s) cubed: V = s³ so take the derivative with respect to time (using the chain rule):

dV/dt = 3(24)²•(0.02) = 34.56 in³/min

If your second question is about the surface area (looks like something got goofed up in typing the question), then repeat this with the surface area equation (Surface area is 6 times the area of one side or SA = 6s²) so d(SA)/dt = 12s•(ds/dt)

Answer 2

V = s^3

dV/dt = 3s^2(ds/dt)

plug in the values s=24, ds/dt = 0.02

dV/dt = 3(24)^2(0.02) = 34.56 cubic inches per minute

another way to do this is

calculate V(24) = 24^3 = 13,824 cubic inches

then calculate V(24.02) = 24.02^3 = 13,858.5

V(24.02)-V(24) = 13,858.5 - 13,824 = 34.58 in^3 (rounding error)

In the 1st method, with derivative of V, you get the instantaneous exact solution, at a point in time

in the 2nd method, you get an approximation, using a small interval of time, of one minute.

but they're close, the same rounded off to one decimal place