Question

A cylinder is to be constructed so that its height plus the perimeter of its base must equal to 2pi cm. Find the ratio of the radius of this cylinder to its height if it is to have a maximum volume.

My work:

SA = 2pirh + 2pir^2
2pi = 2pirh + 2pir^2
0 = 2pirh + 2pir^2 - 2pi
h = (-r^2 + 1) / r


V = pir^2((-r^2 + 1) / r)
V = pi(-r^2 + 1)

And now I don't know what to do? I don't even know if I've done the process in the beginning correctly...


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Text answer is (1 : pi)

Answer 1

You started with surface area, but nowhere is surface area involved.

it said "its height plus the perimeter of its base must equal to 2pi cm"

---> h + 2pir = 2pi
so h = 2pi - (2pi)r

since we want to maximize volume we need a volume equation.

V = pi(r^2)h
= pi(r^2)(2pi - 2pi(r))
= 2(pi)^2(r^2 - r^3)

V' = 3(pi)^2(2r - 3r^2) = 0 for a max of V

it is easy to show that r = 2/3 (r=0 is not possible)

so h = 2pi - 2pi(2/3)
= ....

then find r : h

Answer 2

where I said :
V' = 3(pi)^2(2r - 3r^2) = 0 for a max of V

should obviously have been

V' = <b>2</b>(pi)^2(2r - 3r^2) = 0 for a max of V

Answer 3

2 pi r + h = 2 pi (cm)

Let a = r/h be the unknown ratio

2 pi a*h + h = 2 pi
h (2 pi a + 1) = 2 pi

V = pi r^2 h = pi a^2 h^3
= pi a^2 / [a + (1/2 pi)]^3

Set dV/da = 0 and solve for a.
But check my math first.

Answer 4

If the machine produces 20,000 straws in 8 hours, how many straws can it produce in 50 hours?