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!
To understand why the height of the cylinder must be equal to the diameter of the cross-section to minimize the surface area, let's break down the problem step by step.
Step 1: Define the problem
We need to find the height and diameter of a cylinder that meets two criteria: a specified volume and the minimum surface area.
Step 2: Identify the variables
Let's assign variables to the parameters of the cylinder:
- Let V represent the specified volume.
- Let h represent the height of the cylinder.
- Let d represent the diameter of the cross-section.
Step 3: Express the volume in terms of height and diameter
The volume of a cylinder can be calculated as the product of the cross-sectional area (A) and the height (h). The cross-sectional area, in this case, is the area of a circle, which is πr^2, where r is the radius of the cross-section. Since the diameter is twice the radius, we can represent the radius as d/2. Thus, the volume can be expressed as V = π(d/2)^2h.
Step 4: Express the surface area in terms of height and diameter
The surface area of a cylinder consists of two circles (top and bottom) and one rectangle (side). The area of each circle is πr^2, and the area of the rectangle is the product of the height and the circumference of the cross-section, which is πd. Therefore, the surface area can be expressed as S = 2πr^2 + πdh.
Step 5: Eliminate variables to make a single-variable equation
To find the minimum surface area, we need to express the surface area equation in terms of a single variable. By substituting r with d/2 in the surface area formula, we get S = π(d/2)^2 + πdh.
Step 6: Isolate the single variable and differentiate
To find the minimum value, we can find the critical points by differentiating the surface area equation with respect to the one variable, d. Differentiating S = π(d/2)^2 + πdh gives dS/dd = 4πd/4 + πh = πd + πh.
Step 7: Set the derivative equal to zero
Since we are looking for a critical point, we set dS/dd = 0 and solve for d: πd + πh = 0. This equation confirms that if the height, h, is equal to the diameter, d, then the derivative will be zero.
Step 8: Verify it is a minimum
To confirm that it is a minimum and not a maximum or an inflection point, we can calculate the second derivative, d²S/dd². Since the second derivative is positive for any valid value of d, we can conclude that the surface area is indeed minimized when the height is equal to the diameter.
In conclusion, mathematically proving that the height of the cylinder must be equal to the diameter of the cross-section to minimize the surface area involves expressing the volume and surface area equations in terms of the diameter and height, differentiating the surface area equation with respect to diameter, and confirming that the critical point obtained corresponds to a minimum.