Question

a solid oblique cone with a slant length of 17 units is placed inside an empty cylinder with a congruent base of radius 8 units and a height of 15 units. what is the unfilled volume inside the cylinder? 320π cubic units 597π cubic units 640π cubic units 725π cubic units

Explanation:

Step1: Calculate the volume of the cylinder

The formula for the volume of a cylinder is \(V_{cylinder}=\pi r^{2}h\). Given \(r = 8\) units and \(h=15\) units.

$$V_{cylinder}=\pi\times8^{2}\times15=\pi\times64\times15 = 960\pi$$

Step2: Calculate the volume of the cone

First, find the height of the cone. Using the Pythagorean theorem \(l^{2}=r^{2}+h_{cone}^{2}\), where \(l = 17\) units and \(r = 8\) units. Then \(h_{cone}=\sqrt{l^{2}-r^{2}}=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}=\sqrt{25\times9}=15\) units.
The formula for the volume of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h\). Substituting \(r = 8\) units and \(h = 15\) units, we get \(V_{cone}=\frac{1}{3}\pi\times8^{2}\times15=\frac{1}{3}\pi\times64\times15=320\pi\)

Step3: Calculate the unfilled volume

The unfilled volume \(V=V_{cylinder}-V_{cone}\)

$$V = 960\pi-320\pi=640\pi$$

Answer:

\(640\pi\) cubic units (corresponding to the third option)