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 Question from Heather, a student: A rectangular sponge is increasing its length at 4cm/min, decreasing its width at 2cm/min, and increasing its height at 3cm/min. When its length, width and height are 40, 30, and 20 respectively, find the rate of change of volume and surface area. I'm not quite sure what to do. The third variable is confusing me and I can't seem to set up an equation right. If someone could explain how to set up an equation for this problem, it would be greatly appreciated.

Hi Heather,

The volume of the sponge is given by V = V(t) = LWH where L = L(t) is the length in cm, W = W(t) is the width in cm, H = H(t) is the height in cm and t is the time in minutes. Differentiate both sides of

V(t) = L(t) × W(t) × H(t)

V(t) with respect to t to obtain an expression of the form

V'(t) = an expression involving L(t), W(t), H(t), L'(t), W'(t) and H'(t).

Substitute the values you are given for the right side at the given time to find V'(t) at this time.

Now write the surface area as a function of L, W and H and apply the same technique.

If you have any difficulty completing this just write back and tell us what you have done and we will try to help.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.