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!

Alabaster V.

Thank you.15d