# Implicit differentiation word problem involving the surface area of a cylinder

The radius, $$r$$, of the base of a circular cylinder increases by 2 feet per second while the height, $$h$$, decreases by 1 foot per second. How fast is the surface area of the cylinder changing when the height of the cylinder is $$50$$ feet and the radius of the base is 40 feet? (Hint: The surface area, $$A$$, of such a cylinder is given by $$A = 2πr² + 2πrh$$.)

I was given this question to solve. I took the information that I know: $$\frac{{dr}}{{dt}}$$ = 2 feet per second
$$\frac{{dh}}{{dt}}$$ = −1 foot per second $$r_0 = 40$$ and $$h_0=50$$. Differentiating the area formula gives: $$4\pi r r'+2\pi(r'h+h'r)$$ Plugging it all in gives me: $$A'=4\pi(40)(2)+2\pi(2(50)-1(40))$$Solving that gives $$440\pi$$.
I cannot find an error in my work. However, the answer key gives something else. This is what the answer key says:

This is the correct answer. We are given the following information that is pertinent at the time of interest:
$$\frac{{dr}}{{dt}}$$ = 2 feet per second $$\frac{{dh}}{{dt}}$$ = −1 foot per second
We are interested in the value of $$\frac{{dA}}{{dt}}$$ when $$h = 50$$ feet and $$r = 40$$ feet. To find this, differentiate both sides of the surface area formula implicitly with respect to t and substitute in the given information, as follows: $$\frac{{dA}}{{dt}} = 2π(2r)\frac{{dr}}{{dt}} + 2π(r\frac{{dh}}{{dt}} + h\frac{{dr}}{{dt}})= 4π(40)(2) + 2π(2(−1) + 50(2)) = 320π + 196π = 516π$$
So, the surface area is increasing at a rate of $$516π$$ square feet per second.

The difference I found was in their $$2π(2(−1) + 50(2))$$ were they seem to have a 2 instead of a 1. Am I correct in thinking that they made the error? Or is the error mine? Did I do the implicit differentiation correctly?

• I see the difference as they put for $r=2$ instead of $r=40$. Your implicit differentiation is right – imranfat Aug 6 '19 at 1:00

You did the implicit differentiation correctly. The solution appears to have used 2 in lieu of $$r = 40$$ (for what reason is unbeknownst to me).