The dimensions of a rectangular prism are changing with time. The length is growing at the rate of 1m/s, the width is shrinking at the rate of 2m/s, and the height is growing at the rate of 1m/s.

Question

When the rectangular prism is a cube of side length 10m:

A) how is the volume of the rectangular prism changing?

B) how is the surface area of the rectangular prism changing?

Answer 1

V = lwh

Take derivative with respect to time (product rule)

dV/dt = l [w (dh/dt) + h (dw/t)] + wh [ dl/dt ]

plug in known values

dV/dt = 10[ 10(1) + 10(-2)] + 100 [1]

= 10 [ -10 ] + 100 = 0 so NO change in Volume

SA = 2lw + 2lh + 2hw

Take derivative with respect to time

dSA/dt = 2(l• dw/dt + w• dl/dt) + 2(l• dh/dt + h• dl/dt) + 2(h• dw/dt + w• dh/dt)

plug in known values

dSA/dt = 2(10• -2 + 10• 1) + 2(10• 1 + 10• 1) + 2(10• -2 + 10• 1)

= 2(-10) + 2(20) + 2(-10) = 0 so NO change in Surface Area