Describe in detail what you need	A rectangle with a fixed perimeter of 64 units is rotated about one of its sides, thus sweeping out a figure in the shape of a right circular cylinder. What is the maximum volume of the cylinder? Solution: Let r - be one side of the rectangle h - be the other side
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Maximum Volume of a Cylinder Generated by a Rotated Rectangle

To find the maximum volume of a cylinder generated by rotating a rectangle with a fixed perimeter of 64 units, we first need to establish some relationships between the dimensions of the rectangle and the volume of the resulting cylinder.

Step 1: Define Variables

Let:

• ( r ) = one side of the rectangle (the radius of the cylinder when the rectangle rotates about this side)
• ( h ) = the other side of the rectangle (the height of the cylinder when the rectangle rotates)

Step 2: Set Up the Perimeter Constraint

The perimeter ( P ) of the rectangle is given by the formula: [ P = 2r + 2h ] We know that the perimeter is fixed at 64 units: [ 2r + 2h = 64 ]

By simplifying, we find: [ r + h = 32 ] From this, we can express ( h ) in terms of ( r ): [ h = 32 - r ]

Step 3: Volume of the Cylinder

The volume ( V ) of the cylinder can be calculated using the formula: [ V = \pi r^2 h ] Substituting ( h ) from the equation derived from the perimeter constraint: [ V = \pi r^2 (32 - r) = \pi (32r^2 - r^3) ]

Step 4: Find the Maximum Volume

To maximize the volume, we need to take the derivative of ( V ) with respect to ( r ) and set it to zero: [ \frac{dV}{dr} = \pi (64r - 3r^2) ] Setting the derivative to zero gives us: [ 64r - 3r^2 = 0 ] Factoring out ( r ): [ r(64 - 3r) = 0 ] This provides two possible solutions:

1. ( r = 0 ) (not a valid solution, as it results in no volume)
2. ( 64 - 3r = 0 ) → ( 3r = 64 ) → ( r = \frac{64}{3} )

Now, substituting ( r ) back into the equation for ( h ): [ h = 32 - \frac{64}{3} = \frac{96 - 64}{3} = \frac{32}{3} ]

Step 5: Calculate the Maximum Volume

Now we can plug the values of ( r ) and ( h ) back into the volume equation: [ V = \pi r^2 h = \pi \left(\frac{64}{3}\right)^2 \left(\frac{32}{3}\right) ] Calculating this: [ V = \pi \left(\frac{4096}{9}\right) \left(\frac{32}{3}\right) = \pi \left(\frac{4096 \times 32}{27}\right) = \pi \left(\frac{131072}{27}\right) ]

Conclusion

Thus, the maximum volume of the cylinder generated by the rotation of a rectangle with a fixed perimeter of 64 units is: [ V = \frac{131072\pi}{27} \text{ cubic units.} ]

This represents the largest possible volume that can be obtained under the given constraints, providing an optimum solution for the geometric scenario presented.