height of rect. box is 10 inches. the length inc. at rate of 2 in/sec ,width dec. at a rate 4 in/sec. whenlength is 8 inche and the width is 6 inches, find rate at which volume of box is changing.

Answer 1

I always found optimization problems to be one of the more interesting parts of calculus. Sometimes it might feel like what you're learning has no application but this is not one of those times!

The key in any optimization problem is coming up with a formula for what is being optimized, and then taking the derivative.

Here we are optimizing volume:

V = HLW (height x length x width)

We take the derivative, using the product rule:

dV/dt = dH/dt * LW + dL/dt * HW + dW/dt * HL

Now we just plug in the numbers we are given! (The question doesn't mention rate of change of H so I'm assuming H is constant -> dH/dt = 0)

dV/dt = dL/dt * HW + dW/dt * HL = 2*10*6 - 4*10*8 = 120 - 320 = -200 in/sec

Hope this helps!