An open cylindrical wastepaper bin, of radius r cm and capacity V cm3, is to have a surface area of 5000 cm2.?

Question

"Calculate the maximum possible capacity of the bin" The answer is 38 400 cm3 (3 sf) but I can't get the procedure.

Who can help me with this maths problem?

Answer 1

We know that the surface area of a cylinder is equal to the area of the faces (top and bottom) plus the area of the sides. In this case, this is an open cylinder, so the surface area calculation is as follows:

S = pi*r^2 + 2*pi*r*h
(translation: the area of one of the sides is pi*r^2, since it's a circle, and the surface area of the rest is the circumference of the can times the height).

Therefore, since S = 5000
5000 = pi*r^2 + 2pi*r*h

Let's isolate the h.

5000 - pi*r^2 = 2*pi*r*h
(5000 - pi*r^2)/(2*pi*r) = h

Note that we want to maximize the capacity. Keep in mind that the volume of a cylinder is as follows:

V = pi*r^2*h

But, h = (5000 - pi*r^2)/(2*pi*r)
so we rewrite V in terms of a single variable.

V = pi*r^2* (5000 - pi*r^2)/(2*pi*r)

Note that this is a product with fractions, so we can cancel pi and a single r.

V = r(5000 - pi*r^2)/2

We want to pull out pesky constants, so we pull out the 1/2:

V = 1/2 * r(5000 - pi*r^2)
V = 1/2 * (5000r - pi*r^3)
V = 2500r - (pi/2)r^3

So now we have a function based on one variable, V(r).
V(r) = 2500r - (pi/2)r^3

To get the maximum, we take the derivative, and make it 0.

V'(r) = 2500 - (pi/2)[3r^2]
V'(r) = 2500 - (3pi/2)(r^2)

Now, set it to 0.

0 = 2500 - (3pi/2)(r^2)
(3pi/2)r^2 = 2500
r^2 = 5000/(3pi)
r = sqrt(5000/3pi)

[Note: Normally when taking the square root, we put plus or minus on the right hand side, but given that r, the radius, cannot be negative, we discard the negative result]

To calculate the maximum possible capacity, we plug in the value we just solved for, r, into the Volume function we made.

V(r) = 2500r - (pi/2)r^3
V(sqrt(5000/3pi)) = 2500(sqrt(5000/3pi) - (pi/2)[sqrt(5000/3pi)]^3

From here I'm going to leave it to you to reduce. That should work out to the correct answer, unless I made an error in my steps.

Answer 2

Remember your equations for Volume and surface area (for open cylinder) so that you can optimize the Volume.
SA=(2πrh)+πr^2 and V=hπr^2
Differentiate and solve for r and h.