You are going to make many cylindrical cans. The cans will hold different volumes. But you'd like them all to be such that the amount of sheet metal used for the cans is as small as possible, subject to the can holding the specific volume. How do you choose the ratio of diameter to height of the can? Assume that the thickness of the wall, top, and bottom of the can is everywhere the same, and that you can ignore the material needed for example to join the top to the wall.
Put differently, you ask what ratio of diameter to height will minimize the area of a cylinder with a given volume?
That ratio equals: _______
Oh, nevermind, i figured it out. The answer is 1. It was simpler than i thought
Hmmm. Figured it out in two minutes! Less time than it took you to post the problem, I'd say...
Yes... I had been working on it for some time before i posted it.
To determine the ratio of diameter to height that minimizes the area of a cylinder with a given volume, we can use calculus. Let's break down the problem into steps:
1. Define the variables:
- Let D be the diameter of the cylinder
- Let H be the height of the cylinder
- Let V be the volume of the cylinder
- Let r be the radius of the cylinder, which is equal to half the diameter (r = D/2)
2. Express the volume of the cylinder in terms of D and H:
- The volume of a cylinder is given by V = π * r^2 * H.
- Substituting r = D/2, we get V = π * (D/2)^2 * H = (π/4) * D^2 * H.
3. Express the surface area of the cylinder in terms of D and H:
- The surface area of a cylinder consists of the curved surface area (lateral area) and the areas of the top and bottom.
- The lateral area is given by A_lateral = 2π * r * H.
- The area of each circular end is given by A_end = 2 * π * r^2 (since there are two ends in a cylinder).
- Therefore, the total surface area is A = A_lateral + 2 * A_end = 2π * r * H + 2 * π * r^2.
4. Substitute the expression for r in terms of D into the equation for the surface area:
- A = 2π * (D/2) * H + 2 * π * (D/2)^2 = π * D * H + (π/2) * D^2.
5. Minimize the surface area:
- To minimize the surface area, we need to find the critical points where the derivative of A with respect to D is equal to zero.
- Take the derivative of A with respect to D: dA/dD = π * H + (π/2) * 2D.
- Set dA/dD = 0 and solve for D: π * H + (π/2) * 2D = 0.
- Simplifying the equation, we find D = -2H.
6. Determine the ratio of diameter to height that minimizes the surface area:
- The ratio of diameter to height, D/H, is equal to -2. (Note: The negative sign does not affect the ratio.)
Therefore, the ratio of diameter to height that minimizes the area of a cylinder with a given volume is -2.