Question

A rectangular piece of paper with perimeter 100 cm is to be rolled to form a cylinder tube. Find the dimensions of the paper that will produce a tube with a maximum volume.

Answer 1

To find the dimensions of the paper that will produce a tube with a maximum volume, we need to use optimization techniques.

Step-by-step explanation:

To find the dimensions of the paper that will produce a tube with a maximum volume, we need to use optimization techniques. Let the length of the rectangle be L and the width be W.

The perimeter of the rectangle is given as 100 cm, so, 2L + 2W = 100. We need to express the volume of the cylinder tube in terms of a single variable to optimize it, which can be done by relating the height of the cylinder tube to the length of the rectangle.

The formula for the volume of a cylinder is V = πr^2h, where r is the radius of the base and h is the height. Since the height of the cylinder tube is equal to the length of the rectangle, h = L. The radius of the base can be found by dividing the width of the rectangle by 2, r = W/2.

Substituting the values of r and h into the volume formula, we have V = π(W/2)^2L = πW^2L/4. To optimize the volume, we need to maximize V with respect to a single variable, which is W in this case.

1. Using the perimeter equation, solve for W in terms of L: W = 50 - L.
2. Substitute W into the volume equation: V = π(50 - L)^2L/4.
3. To find the maximum volume, take the derivative of V with respect to L and set it equal to zero. Solve for L to find the value that maximizes V.
4. Once you have the value of L, substitute it back into the perimeter equation to find W.

Finally, substitute the values of L and W into the volume equation to find the maximum volume of the cylinder tube.