Hi 39mello. Are you having trouble getting started?
Again, the inside volume of the cup is a cylinder. Let's think about that cylinder's surface area (no top). Two areas comprise that surface: one is circular (the base), and the other is rectangular (the wall).
Let's define the variables needed to express those areas:
r = the cylinder's radius
h = the cylinder's height
The area of the circular base is expressed as Pi*r^2.
If we mentally unroll a cylinder's wall, we see that it's a rectangle. The width of that rectangle is the same as the circumference of the circular base, and its height is h. The circumference of a circle is 2*Pi*r. So, the area of the cylinder's wall (width×height) is expressed as 2*Pi*r*h.
Therefore, a formula for our no-top cylinder's total surface area (A) is:
A = Pi*r^2 + 2*Pi*r*h
We need an expression for h in terms of r, for the volume function. We can obtain that expression, by substituting the given area into the formula above and solving for h.
48*Pi = Pi*r^2 + 2*Pi*r*h
Have a go at it, and post what you get for h. If you're able to write the cylinder's volume function V(r), then please share that, too. Cheers!
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