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A solid oblique cone with a slant height 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?

A. [tex]$320 \pi$[/tex] cubic units
B. [tex]$597 \pi$[/tex] cubic units
C. [tex]$640 \pi$[/tex] cubic units
D. [tex]$725 \pi$[/tex] cubic units


Sagot :

Let's solve the problem step-by-step.

1. Understanding the problem:
- We have a cylinder with:
- Radius \( r = 8 \) units
- Height \( h = 15 \) units
- Inside this cylinder, there is an oblique cone with:
- The same radius of the base \( r = 8 \) units
- Slant height \( l = 17 \) units

2. Volume of the cylinder:
- The formula for the volume of a cylinder is:
[tex]\[ V_{\text{cylinder}} = \pi r^2 h \][/tex]
- Substituting the values:
[tex]\[ V_{\text{cylinder}} = \pi (8)^2 (15) = \pi \times 64 \times 15 = 960 \pi \quad \text{cubic units} \][/tex]

3. Determining the height of the cone:
- We use the Pythagorean theorem for the triangle formed by the radius, height, and slant height of the cone:
[tex]\[ l^2 = r^2 + h_{\text{cone}}^2 \][/tex]
[tex]\[ 17^2 = 8^2 + h_{\text{cone}}^2 \][/tex]
[tex]\[ 289 = 64 + h_{\text{cone}}^2 \][/tex]
[tex]\[ h_{\text{cone}}^2 = 225 \][/tex]
[tex]\[ h_{\text{cone}} = \sqrt{225} = 15 \quad \text{units} \][/tex]

4. Volume of the cone:
- The formula for the volume of a cone is:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}} \][/tex]
- Substituting the values:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi (8)^2 (15) = \frac{1}{3} \pi \times 64 \times 15 = \frac{960 \pi}{3} = 320 \pi \quad \text{cubic units} \][/tex]

5. Unfilled volume inside the cylinder:
- The unfilled volume is the volume of the cylinder minus the volume of the cone:
[tex]\[ V_{\text{unfilled}} = V_{\text{cylinder}} - V_{\text{cone}} \][/tex]
[tex]\[ V_{\text{unfilled}} = 960 \pi - 320 \pi = 640 \pi \quad \text{cubic units} \][/tex]

The correct answer is:
[tex]\[ 640 \pi \text{ cubic units} \][/tex]
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