A right circular cylinder with no top has a volume of 50 cubic meters. Find the radius that will yield minimum surface area. Let x = radius.

Question

Hi, I have been stuck on this rational function question for hours. I tried plugging in 50 for the volume in the volume of a cylinder formula but I still can't seem to find the answer. Please help.

Answer 1

Anthony G. has a very nice solution. However, if you understand derivatives and/or you can't or don'y know how to graph the equation, you can take the first derivative, set it equal to zero, and solve for r or "x".

S=∏r^2+100/r

S=∏x^2+100x^(-1)

S'=2∏x-100x^(-2)

S'=2∏x-100/x^2

find a common denominator, which is x^2

multiply both terms of 2∏x by x^2

S'=2∏x^3/x^2-100/x^2

S'=(2∏x^3-100)/x^2

0=(2∏x^3-100)/x^2

the only way the right side becomes zero is if the numerator becomes zero

0=2∏x^3-100

0=∏x^3-50

50=∏x^3

50/∏=x^3

15.915494=x^3

take the cube root of both sides

x=(15.915494)^(1/3)

x=2.51539 m is the radius as you can see above

Answer 2

Hi Alabaster! This is definitely a tricky one, but we can figure it out!

Let's start by listing what we know about the cylinder. We know the formula and value for the volume:

V = 50

V = π r^2 h

We don't know the value of the surface area, but we do know the formula (remember to use the one for a cylinder with no top):

S = 2πrh + π r^2

Now that we have everything we know written down, we can begin solving the problem. We are told that the radius is x, so our formulas can be rewritten as:

V = π x^2 h

S = 2πxh + π x^2

You mentioned that you plugged in the volume's value of 50. That's great! Let's do that:

50 = π x^2 h

Now, here's the part where we can get stuck... what next? Well, the "h" is really annoying and ruins our nice equations. I wish there was just an x... aha! I can solve the volume equation for h:

50/(π x^2) = h

We're getting somewhere... Let's plug this expression for h into our surface area formula:

S = 2πx * 50/(π x^2) + π x^2

Simplifying we get:

S = 100/x + π x^2

Alright... so... where are we now? Well, we have a formula for the surface area that has only the single variable we wanted: x. We are ready to find the value of x that will yield the minimum (smallest) surface area.

Depending on your class/teacher, you may have been taught a specific method to use to solve the problem from this point. In my Precalculus class, we had to graph the equation (using something like Desmos) to find the minimum value visually.

So, graphing 100/x + π x^2, and zooming out so we can see around 80 on the x and y-axis: I can see that there is a low point in the curve somewhere between 0 and 4. Zooming in closer (or clicking on the low point), we find that the coordinate is approximately (2.515, 59.633).

The radius that will yield the minimum surface area is the x-value: 2.515 meters.

Hope that helps! Good luck!