Question

challenge find the volume of the solid where the cone and half sphere are hollow. use 3.14 for π. the volume of the solid is about \\(\square\\) in.\\(^3\\). (round to the nearest whole number as needed.)

Explanation:

Step1: Identify the shapes (cylinder, cone, hemisphere)

The solid is a cylinder with a hollow cone and a hollow hemisphere. Radius \( r = 9 \) in, height of cylinder \( h_{cyl} = 34 \) in, height of cone \( h_{cone}=9 \) in (since cone's height equals radius? Wait, no—wait, the cone is inside the cylinder? Wait, the diagram: the cylinder has length 34 in, radius 9 in. The cone has radius 9 in, height equal to the radius? Wait, no, maybe the cone's height is 9? Wait, no, let's recall formulas.

Volume of cylinder: \( V_{cyl} = \pi r^2 h \)

Volume of cone: \( V_{cone} = \frac{1}{3}\pi r^2 h \)

Volume of hemisphere: \( V_{hemisphere} = \frac{2}{3}\pi r^3 \)

Given \( r = 9 \) in, \( h_{cyl} = 34 \) in, \( h_{cone} = 9 \) in (assuming the cone's height is 9, since the radius is 9 and the diagram shows the cone's height as 9? Wait, the diagram: the cylinder length is 34, radius 9. The cone is inside, with radius 9, height 9? Wait, no, maybe the cone's height is 9? Wait, let's check the numbers.

First, calculate volume of cylinder: \( V_{cyl} = 3.14 \times 9^2 \times 34 \)

Calculate \( 9^2 = 81 \), so \( 3.14 \times 81 \times 34 \). Let's compute that: 3.14*81=254.34; 254.34*34=8647.56

Volume of cone: \( V_{cone} = \frac{1}{3} \times 3.14 \times 9^2 \times 9 \) (since cone's height is 9, same as radius? Wait, maybe the cone's height is 9. So \( 9^2=81 \), \( 81 \times 9 = 729 \), \( \frac{1}{3} \times 3.14 \times 729 = 3.14 \times 243 = 763.02 \)

Volume of hemisphere: \( V_{hemisphere} = \frac{2}{3} \times 3.14 \times 9^3 \). \( 9^3 = 729 \), \( \frac{2}{3} \times 3.14 \times 729 = 3.14 \times 486 = 1526.04 \)

Now, the solid's volume is \( V_{cyl} - V_{cone} - V_{hemisphere} \)

So subtract: 8647.56 - 763.02 - 1526.04

First, 8647.56 - 763.02 = 7884.54

Then, 7884.54 - 1526.04 = 6358.5

Wait, that can't be right. Wait, maybe the cone's height is 34? No, the diagram shows the cone's height as 9? Wait, maybe I misread the diagram. Wait, the diagram: the cylinder has length 34, radius 9. The cone is inside, with radius 9, and the hemisphere is also inside. Wait, maybe the cone's height is 9, and the hemisphere is attached? Wait, no, the problem says "the cone and half sphere are hollow". So the solid is a cylinder, with a hollow cone and a hollow hemisphere. So we need to subtract the volume of the cone and the volume of the hemisphere from the cylinder's volume.

Wait, let's recheck the formulas:

Cylinder: \( V = \pi r^2 h \), \( r=9 \), \( h=34 \): \( 3.14 * 81 * 34 = 3.14 * 2754 = 8647.56 \) (since 81*34=2754)

Cone: \( V = (1/3)\pi r^2 h \), \( r=9 \), \( h=9 \) (assuming the cone's height is 9, as per the diagram's 9 in label). So \( (1/3)*3.14*81*9 = 3.14*81*3 = 3.14*243 = 763.02 \)

Hemisphere: \( V = (2/3)\pi r^3 \), \( r=9 \): \( (2/3)*3.14*729 = 3.14*486 = 1526.04 \)

Now, subtract cone and hemisphere from cylinder: \( 8647.56 - 763.02 - 1526.04 = 8647.56 - (763.02 + 1526.04) = 8647.56 - 2289.06 = 6358.5 \). Rounded to nearest whole number: 6359? Wait, but 6358.5 rounds to 6359? Wait, but maybe I made a mistake in the cone's height. Wait, maybe the cone's height is 34? No, the diagram shows the 9 in as the radius, and the cone's height is 9? Wait, no, maybe the cone's height is 9, and the hemisphere is a half-sphere with radius 9. Let's check again.

Wait, maybe the cone's height is 9, and the cylinder's height is 34. So:

Cylinder: 3.14 * 9^2 * 34 = 3.14 * 81 * 34 = 3.14 * 2754 = 8647.56

Cone: (1/3)*3.14*9^2*9 = (1/3)*3.14*81*9 = 3.14*243 = 763.02

Hemisphere: (2/3)*3.14*9^3 = (2/3)*3.14*729 = 3.14*486 = 1526.04

Total hollow volume: 763.02 + 1526.04 = 2289.06

Subtract from cylinder: 8647.56 - 2289.06 = 6358.5 ≈ 6359

Wait, but maybe the cone's height is 34? No, the diagram shows the 9 in as the radius, and the cone's height is 9. Wait, maybe I misread the diagram. Let me check the problem again: "the cone and half sphere are hollow". The diagram: a cylinder with length 34, radius 9, a cone inside (with radius 9, height 9) and a hemisphere (radius 9) at the end. So yes, that makes sense. So the calculations seem right.

Wait, but let's recalculate:

Cylinder: 3.14 * 9 * 9 * 34 = 3.14 * 81 * 34. 81*34: 80*34=2720, 1*34=34, so 2720+34=2754. 3.14*2754: 2754*3=8262, 2754*0.14=385.56, so 8262+385.56=8647.56. Correct.

Cone: (1/3)*3.14*9*9*9. 9*9*9=729. (1/3)*729=243. 3.14*243=763.02. Correct.

Hemisphere: (2/3)*3.14*9*9*9. 9*9*9=729. (2/3)*729=486. 3.14*486=1526.04. Correct.

Subtract: 8647.56 - 763.02 = 7884.54; 7884.54 - 1526.04 = 6358.5. Rounded to nearest whole number: 6359.

Wait, but maybe the cone's height is 34? Let's try that. If cone's height is 34:

V_cone = (1/3)*3.14*9^2*34 = (1/3)*3.14*81*34 = 3.14*27*34 = 3.14*918 = 2882.52

Then V_hemisphere=1526.04

Total hollow: 2882.52 + 1526.04 = 4408.56

Subtract from cylinder: 8647.56 - 4408.56 = 4239. But that doesn't match the diagram. The diagram shows the cone's height as 9, since the radius is 9 and the cone's height is equal to the radius? Maybe. So the first calculation is correct. So the volume is approximately 6359.

Wait, but let's check the problem statement again. The diagram: "34 in" is the length of the cylinder, "9 in" is the radius. The cone is inside, with radius 9, and height 9 (since the diagram shows the cone's height as 9). The hemisphere is also radius 9. So yes, the first calculation is correct.

Answer:

6359