Question

at a certain instant the dimensions of a rectangular parallelepiped are 4,5,&6ft. and they are increasing respectively at the rates of 1,2,&3ft per second. at what rate is the volume changing?

Answer 1

V = LWH

You need the product rule to do this one. However, you must break it into two groups.

dV/dt = d/dt[LW]*H+LW*d/dt[h]
dV/dt = [dL/dt*W+L*dW/dt]H+LW*dH/dt
dV/dt = WH*dL/dt + LH*dW/dt + LW*dH/dt
L = 4; W = 5; H = 6; dL/dt = 1; dW/dt = 2; dH/dt = 3

dV/dt = 5*6*1+4*6*2+4*5*3 = 138 ft^3/s