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 solve this problem, let's break it down into smaller steps:
Step 1: Define the problem
We are given that we need to make a cylinder with a specified volume. We want to find the dimensions (height and diameter) of the cylinder that minimize its surface area.
Step 2: Formulate the problem mathematically
Let's denote the height of the cylinder as h and the diameter of the cross-section as d. We know that the volume of a cylinder is given by:
Volume = π * (d/2)^2 * h
And the surface area of a cylinder is given by:
Surface Area = 2π * (d/2) * h + π * (d/2)^2
Step 3: Express the surface area in terms of a single variable
To express the surface area in terms of a single variable, we can use the volume equation and solve it for h:
h = (4 * Volume) / (π * d^2)
Step 4: Substitute the value of h into the surface area equation
Substituting the value of h into the surface area equation, we get:
Surface Area = 2π * (d/2) * [(4 * Volume) / (π * d^2)] + π * (d/2)^2
Simplifying this expression, we have:
Surface Area = 2 * (2 * Volume / d) + π * (d/2)^2
Surface Area = 4 * Volume / d + π * (d/2)^2
Step 5: Find the minimum surface area
To find the minimum surface area, we need to take the derivative of the surface area equation with respect to d and set it equal to zero:
d(Surface Area) / dd = -4 * Volume / d^2 + 2 * π * (d/2) = 0
Simplifying this equation, we have:
-4 * Volume / d^2 + π * d = 0
Cross-multiplying, we get:
-4 * Volume + π * d^3 = 0
Simplifying further:
π * d^3 = 4 * Volume
d^3 = (4 * Volume) / π
d = [(4 * Volume) / π]^(1/3)
Step 6: Find the height, h
Substitute the value of d back into the equation we derived in step 3:
h = (4 * Volume) / (π * d^2)
h = (4 * Volume) / (π * [(4 * Volume) / π]^(2/3))
Step 7: Simplify the height equation
Simplifying the height equation, we get:
h = (4 * Volume) / (4 * Volume)^(2/3)
h = (4 * Volume)^(1/3)
Step 8: Conclusion
From step 6, we can see that the height of the cylinder, h, is equal to the diameter, d. Therefore, if we want to minimize the surface area of the cylinder while maintaining a specified volume, the height of the cylinder must be equal to the diameter of the cross-section.
By following these steps, we have mathematically proven that the height of the cylinder must be equal to the diameter of the cross-section in order to minimize its surface area.