Problem number six states, a rectangular sheet of paper has a length of x and a width of y, where x is greater than y. The rectangle may be folded to form the curved surface of a circular cylinder in two different ways. What is the ratio of the volume of the longer cylinder to the volume of the shorter cylinder? So let's start out with our rectangular piece of paper. We know that the longer side is x and that the shorter side is y. We can make two cylinders out of this piece of paper depending on which way we fold it. For the first combination the circumference would be x and the height would be y, while for the second combination the circumference would be y and the height x. Let's try and solve for the radii, we'll call them r1 and r2. In this case let's represent them as the circumference since the circumference of each cylinder is known, so circumference is diameter times pi and we know the diameter is twice the radius, so we get an expression of 2r1 times pi equals x and 2r2 times pi equals y. Solving for r1 and r2 by dividing both sides by 2pi, we get x over 2pi and y over 2pi. Now that we know the radius, let's solve for the volume. The volume of a cylinder is pi times the radius squared times the height. So for the first cylinder, pi times r1 squared times y, and for the second cylinder pi times r2 squared times x. Now let's substitute our radius equations into the volume equation, simplifying this is what we end up getting. Now let's cancel out the pi in the numerator and the denominator and we get x squared over 4pi y and y squared over 4pi x. We know that x is greater than y, so we know that v2 will be the volume of the longer cylinder. So the question asked for the ratio of v2 to v1, which is the longer cylinder to the shorter cylinder. Let's substitute in the equations, and cancel out the denominators on both sides, and finally divide y and x on both sides to get a final ratio of y to x, and this is our final answer. So the question asked us, what is the ratio of the volume of the longer cylinder to the volume of the shorter cylinder? The answer is y to x, letter B.

cylinder

volume ratio

paper folding

rectangular sheet

geometry