Let me analyze the image and solve the problems step-by-step.

First Problem:

We are tasked to find the volume of the piece of wood, where the central cone is cut out of a cylinder.

Given:

• Cylinder:
  • Radius $r = 6 \, \text{cm}$
  • Height $h = 12 \, \text{cm}$
• Cone:
  • Radius $r = 6 \, \text{cm}$ (same as the cylinder's radius)
  • Height $h = 12 \, \text{cm}$ (same as the cylinder's height)

Step 1: Calculate the volume of the cylinder:

The volume of a cylinder is given by: $V_{\text{cylinder}} = \pi r^2 h$ Substitute $r = 6$ and $h = 12$: $V_{\text{cylinder}} = \pi (6)^2 (12) = \pi (36)(12) = 432\pi \, \text{cm}^3$

Step 2: Calculate the volume of the cone:

The volume of a cone is given by: $V_{\text{cone}} = \frac{1}{3} \pi r^2 h$ Substitute $r = 6$ and $h = 12$: $V_{\text{cone}} = \frac{1}{3} \pi (6)^2 (12) = \frac{1}{3} \pi (36)(12) = 144\pi \, \text{cm}^3$

Step 3: Subtract the cone's volume from the cylinder's volume:

$V_{\text{wood}} = V_{\text{cylinder}} - V_{\text{cone}}$ $V_{\text{wood}} = 432\pi - 144\pi = 288\pi \, \text{cm}^3$

Final Answer (approximation):

$V_{\text{wood}} = 288\pi \approx 904 \, \text{cm}^3 \, \text{(rounded to the nearest whole number)}.$

Second Problem:

A cylinder and a cone have the same volume. We need to find the height of the cone.

Given:

• Cylinder:
  • Radius $r = 2 \, \text{inches}$
  • Height $h = 27 \, \text{inches}$
• Cone:
  • Radius $r = 3 \, \text{inches}$
  • Height $h \, \text{(to be determined)}$

Step 1: Find the volume of the cylinder:

The volume of a cylinder is given by: $V_{\text{cylinder}} = \pi r^2 h$ Substitute $r = 2$ and $h = 27$: $V_{\text{cylinder}} = \pi (2)^2 (27) = \pi (4)(27) = 108\pi \, \text{in}^3$

Step 2: Set the cone's volume equal to the cylinder's volume:

The volume of a cone is: $V_{\text{cone}} = \frac{1}{3} \pi r^2 h$ Substitute $r = 3$ and equate it to $108\pi$: $\frac{1}{3} \pi (3)^2 h = 108\pi$ $\frac{1}{3} \pi (9) h = 108\pi$ $3h = 108$ $h = \frac{108}{3} = 36 \, \text{inches}$

Final Answer:

The height of the cone is: $h = 36 \, \text{inches}.$

Summary of Answers:

1. Volume of the piece of wood: $904 \, \text{cm}^3$ (rounded to the nearest whole number).
2. Height of the cone: $36 \, \text{inches}$.

Do you have any questions or need further clarification?

5 Relative Questions:

1. How do you calculate the volume of a cylinder?
2. What is the difference in formula between the volume of a cylinder and a cone?
3. Why is the volume of a cone one-third that of a cylinder with the same base and height?
4. How does changing the radius affect the volume of a cone or cylinder?
5. What are real-world applications of calculating volumes of solids?

One Tip:

Always ensure units are consistent throughout a calculation; converting units when needed helps avoid mistakes!