A cylinder is to be made of circular cross-section with a specified volume. Prove that if the surface area is to be a minimum, then the height of the cylinder must be equal to the diameter of the cross-section of the cylinder.
Maybe it's the wording, but I have not been able to crack this one for the past half-hour!
Patience,
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To prove that the height of a cylinder must be equal to the diameter of its circular cross-section for the surface area to be a minimum, we can use calculus. Let's break down the problem step by step:
1. Consider a cylinder with radius 'r' and height 'h'. The volume (V) of this cylinder is given by V = πr^2h.
2. To find the surface area (A) of the cylinder, we need to add the areas of the two circular bases and the lateral surface area. The lateral surface area is the curved surface area of a rectangle when the rectangle is unrolled, given by A = 2πrh.
3. Now we have two equations defining the volume and surface area:
V = πr^2h (Equation 1)
A = 2πrh (Equation 2)
4. We are given that the volume is fixed, so let's differentiate Equation 2 with respect to 'h' since we are trying to find the minimum surface area. This will help us find the critical point where the derivative equals zero and confirm if it is a minimum point.
dA/dh = 2πr (Equation 3)
5. Equation 3 tells us that the derivative of the surface area with respect to 'h' is a constant value, '2πr', which implies the surface area is not dependent on 'h'.
6. Since the surface area is not dependent on the cylinder's height, it means that changing the height will not affect the surface area. Thus, to minimize the surface area, the height needs to be as small as possible.
7. Now, let's consider what happens if the height is increased to a value greater than the radius. In this case, the cylinder would look more like a drum, and the surface area would increase. Thus, the minimum surface area is obtained when the height is equal to the diameter.
Hence, we have proven that if the surface area of a cylinder is to be a minimum, the height of the cylinder must be equal to the diameter of its circular cross-section.