A cylinder and a cone start with the same radius and height. The radius of the cone is then tripled, and the height of the cone is cut in half. The radius of the cylinder stays the same, but the height of the cylinder is doubled. Which change produces a greater increase in volume (i.e., which figure’s volume increases by a larger factor)? Justify your answer. Write “pi” for and “r^2” for .

Answer:

Reponses may vary but should include some or all of the following information: The original volume of the cone is V= (1/3)pi(r^2)h. Tripling the radius (3r) and halving the height (h/2), we get a new volume of V=(1/3)pi(3r)^2(h/2), or V=(3/2)pi(r^2)h. Thus, the cone’s volume has increased by a factor of (3/2)/(1/3) = 9/2, or 4.5. The original volume of the cylinder is V=pi(r^2)h. Doubling the height, we get a new volume of V=pi(r^2)(2h) = 2pi(r^2)h. Thus, the cylinder’s volume has increased by a factor of 2/1, or 2. Therefore, the changes produced a greater increase in volume in the cone than in the cylinder.