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 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, so $V_{cylinder}=\pi\times8^{2}\times15=\pi\times64\times15 = 960\pi$ cubic - units.

Step2: Calculate volume of the cone

First, find the height of the cone using the Pythagorean theorem. The slant - height $l = 17$ units and the radius $r = 8$ units. Let the height of the cone be $h_{cone}$. Then $h_{cone}=\sqrt{l^{2}-r^{2}}=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}=\sqrt{25\times9}=\sqrt{225}=15$ units. The formula for the volume of a cone is $V_{cone}=\frac{1}{3}\pi r^{2}h_{cone}$. Substituting $r = 8$ units and $h_{cone}=15$ units, we get $V_{cone}=\frac{1}{3}\pi\times8^{2}\times15=\pi\times64\times5 = 320\pi$ cubic - units.

Step3: Calculate the unfilled volume

The unfilled volume $V = V_{cylinder}-V_{cone}$. Substitute $V_{cylinder}=960\pi$ and $V_{cone}=320\pi$ into the formula. So $V=960\pi - 320\pi=640\pi$ cubic - units.

Answer:

640π cubic units