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: Recall 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 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 \(V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi(\frac{d}{2})^{2}h=\frac{1}{12}\pi d^{2}h\) and still calculate the volume using the known diameter \(d\) and height \(h\).

Step4: Analyze option A

If we know the radius \(r\) and circumference \(C = 2\pi r\) (but circumference gives no new information about calculating the volume as we already have the radius from \(C = 2\pi r\) and we still lack the height).

Step5: Analyze option D

If we know the diameter \(d\) (so \(r=\frac{d}{2}\)) and slant height \(l\), we need to use the Pythagorean theorem \(h=\sqrt{l^{2}-r^{2}}\) to find the height first (assuming we know it's a right - circular cone). But the problem is asking what is "must be known" directly. We need an additional step (using the Pythagorean theorem) if we know diameter and slant height, while for radius/ diameter and height, we can directly use the volume formula without an extra step related to finding the height from other non - height related measurements.

Answer:

B. Radius and height, C. Diameter and height