Question

a right cone is placed in an empty right cylinder.

• the cone and the cylinder have the same height.
• the circular bases of the cone and the cylinder have the same diameter.
• the volume of the cone is 9 inches cubed.

what is true about the volume of space in the cylinder surrounding the cone?
complete the sentences by selecting the correct answers from the drop-down menus.
the volume of space in the cylinder surrounding the cone is choose... as great as the volume of the cone.
the volume of the space in the cylinder surrounding the cone choose... es cubed.
(dropdown options for first part: one-third, two-thirds, two times, three times)

Explanation:

Step1: Recall volume formulas

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

Step2: Analyze relationship

Since cone and cylinder have same $r$ (same diameter) and same $h$, $V_{cylinder} = 3V_{cone}$ (because $V_{cylinder}=\pi r^2 h = 3\times\frac{1}{3}\pi r^2 h = 3V_{cone}$).

Step3: Find surrounding volume

Volume surrounding cone = $V_{cylinder} - V_{cone} = 3V_{cone} - V_{cone} = 2V_{cone}$. So it's 2 times the cone's volume.

Step4: Calculate numerical value

Given $V_{cone}=9$ in³, surrounding volume = $2\times9 = 18$ in³. But first part: ratio is 2 times.

Answer:

s:
First drop - down: two times
Second drop - down: 18 (Note: For the first blank, from the analysis, the volume of the space in the cylinder surrounding the cone is two times as great as the volume of the cone. For the second blank, since the volume of the cone is 9 cubic inches, the surrounding volume is $2\times9 = 18$ cubic inches.)