Question

which of the following best explains why a cones volume is \\( \frac { 1 } { 3 } \\) that of a cylinder?
a. a cone occupies one - third of the space of a cylinder with the same base and height.
b. a cone has a smaller base area than a cylinder.
c. a cone is shorter than a cylinder with the same radius and height.
d. a cones height is always one - third of its radius.
what happens to the volume of a cone if the height is tripled while the radius remains constant?
a. the volume remains unchanged
b. the volume is tripled
c. the volume is doubled
d. the volume is halved

Explanation:

Step1: Analyze the first question

The volume formula of a cylinder is \(V_{cylinder}=\pi r^{2}h\) and the volume formula of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). When a cone and a cylinder have the same base (\(r\) is the same, so base - area \(A = \pi r^{2}\) is the same) and height (\(h\) is the same), the cone's volume is one - third of the cylinder's volume. Option b is wrong because if they have the same radius, they have the same base area. Option c is wrong because when comparing the volume formula relationship, we assume the same height. Option d is wrong as there is no such general relationship between a cone's height and radius for volume comparison with a cylinder.

Step2: Analyze the second question

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\). If the radius \(r\) remains constant and the height \(h\) is tripled (\(h' = 3h\)), then the new volume \(V'=\frac{1}{3}\pi r^{2}(3h)\). Using the commutative property of multiplication \(V' = 3\times\frac{1}{3}\pi r^{2}h\). Since \(\frac{1}{3}\pi r^{2}h=V\), we have \(V' = 3V\).

Answer:

For the first question: a. A cone occupies one - third of the space of a cylinder with the same base and height.
For the second question: b. The volume is tripled.