Question

how can the volume of a cone be described compared to a cylinder with the same base and height?
a. it is twice the volume
b. it is the same volume
c. it is one - third the volume
d. it is half the volume
how does the volume of a cone change if the radius is tripled?
a. the volume doubles
b. the volume triples
c. the volume increases by a factor of 6
d. the volume increases by a factor of 9
which of the following units would be appropriate for measuring the volume of a cone?
a. cubic centimeters
b. meters
c. square centimeters
d. centimeters
if the radius of a cone is halved, how is the volume affected?
a. the volume remains the same
b. the volume is halved
c. the volume is doubled
d. the volume is reduced to one - fourth

Explanation:

Step1: Compare cone and cylinder volumes

Volume of cylinder: $V_{cyl} = \pi r^2 h$
Volume of cone: $V_{cone} = \frac{1}{3}\pi r^2 h$
Thus $V_{cone} = \frac{1}{3}V_{cyl}$

Step2: Effect of tripling cone radius

Original volume: $V = \frac{1}{3}\pi r^2 h$
New volume ($r'=3r$): $V' = \frac{1}{3}\pi (3r)^2 h = 9\times\frac{1}{3}\pi r^2 h = 9V$

Step3: Identify volume units

Volume uses 3-dimensional (cubic) units; linear units (meters, centimeters) measure length, square units measure area.

Step4: Effect of halving cone radius

Original volume: $V = \frac{1}{3}\pi r^2 h$
New volume ($r'=\frac{1}{2}r$): $V' = \frac{1}{3}\pi (\frac{1}{2}r)^2 h = \frac{1}{4}\times\frac{1}{3}\pi r^2 h = \frac{1}{4}V$

Answer:

1. c. It is one-third the volume
2. d. The volume increases by a factor of 9
3. a. Cubic centimeters
4. d. The volume is reduced to one-fourth