Final answer:
To find the length of the radius that maximizes the volume of the cylinder, use the formula V = πr²h and the given surface area of the cylinder to determine the height in terms of the radius. Substitute the expression for the height into the volume formula, and then find the value of the radius that satisfies the condition dV/dr = 0. The length of the radius that maximizes the volume of the cylinder is √(22/π) inches.
Step-by-step explanation:
To find the length of the radius that maximizes the volume of the cylinder, we first need to determine the formula for the volume of a cylinder. The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.
In this case, we are given that the cylinder is made from 66 square inches of material. To maximize the volume, we need to find the length of the radius that will use up all of the given material.
Since the surface area of a cylinder can be calculated by adding the areas of the two circular bases and the area of the curved side, we have:
- 2πrh + πr² = 66 square inches
- 2πrh = 66 - πr²
- h = (66 - πr²) / (2πr)
Now, we can substitute the expression for h into the formula for the volume:
- V = πr²((66 - πr²)/(2πr))
- V = (33r - (πr³)/2)
To maximize the volume, we need to find the value of r that satisfies the condition dV/dr = 0.
- dV/dr = 33 - (3πr²)/2 = 0
- 33 = (3πr²)/2
- r² = (2*33)/(3π)
- r² = 22/(π)
- r = √(22/π)
Therefore, the length of the radius that maximizes the volume of the cylinder is √(22/π) inches.