Question

1. what must be known to calculate the volume of a cone?

a. radius and circumference
b. radius and height
c. diameter and height
d. diameter and slant height

Explanation:

Step1: Write the volume formula of a cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\), where \(r\) is the radius of the base of the cone and \(h\) is the height of the cone.

Step2: Analyze option B

If we know the radius \(r\) and height \(h\), we can directly substitute them into the formula \(V = \frac{1}{3}\pi r^{2}h\) to calculate the volume.

Step3: Analyze option C

If we know the diameter \(d\), since \(r=\frac{d}{2}\), we can substitute \(r=\frac{d}{2}\) into the volume formula \(V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi(\frac{d}{2})^{2}h=\frac{1}{12}\pi d^{2}h\). So if we know the diameter \(d\) and height \(h\), we can also calculate the volume.

Step4: Analyze option A

If we know the radius \(r\) and circumference \(C = 2\pi r\) (circumference is redundant information as we already know \(r\) from \(C = 2\pi r\)), but this still doesn't give us the height \(h\) needed for the volume formula.

Step5: Analyze option D

If we know the diameter \(d\) (or radius \(r=\frac{d}{2}\)) and slant height \(l\), we can find the height \(h=\sqrt{l^{2}-r^{2}}\) using the Pythagorean theorem (\(l^{2}=h^{2}+r^{2}\) for a right - circular cone). But this requires an extra step of calculating \(h\) from \(l\) and \(r\) (or \(d\)), while options B and C give the necessary values (either \(r\) or can be easily converted to \(r\) and \(h\) directly) for the volume formula without an extra calculation step. Since the question is about what must be known (the most straightforward information), B and C are more appropriate.

Answer:

B. Radius and height, C. Diameter and height