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 cylinder volume

Volume formula: $V_{cyl} = \pi r^2 h$
Substitute $r=8, h=15$:
$V_{cyl} = \pi \times 8^2 \times 15 = \pi \times 64 \times 15 = 960\pi$

Step2: Find cone height

Use Pythagorean theorem: $h_{cone} = \sqrt{l^2 - r^2}$
Substitute $l=17, r=8$:
$h_{cone} = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15$

Step3: Calculate cone volume

Volume formula: $V_{cone} = \frac{1}{3}\pi r^2 h$
Substitute $r=8, h=15$:
$V_{cone} = \frac{1}{3} \times \pi \times 8^2 \times 15 = \frac{1}{3} \times \pi \times 64 \times 15 = 320\pi$

Step4: Find unfilled volume

Subtract cone from cylinder:
$V_{unfilled} = V_{cyl} - V_{cone} = 960\pi - 320\pi$

Answer:

$640\pi$ cubic units