a toy consists of a cylinder of diameter 6cm sandwiched between a hemisphere and a cone of the same diameter. if the cone is of height 8cm and the cylinder is of height 10cm, find the total volume of the toy. 

To find the total volume of the toy, we need to calculate the volumes of the three individual components: the hemisphere, the cylinder, and the cone. Then, we'll sum these volumes.

### 1. Volume of the Hemisphere
The volume \( V \) of a hemisphere with radius \( r \) is given by:

\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\]

Given:
- Diameter of the hemisphere = 6 cm, so the radius \( r = \frac{6}{2} = 3 \) cm.

\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (3)^3 = \frac{2}{3} \pi \times 27 = 18\pi \text{ cm}^3
\]

### 2. Volume of the Cylinder
The volume \( V \) of a cylinder with radius \( r \) and height \( h \) is given by:

\[
V_{\text{cylinder}} = \pi r^2 h
\]

Given:
- Radius \( r = 3 \) cm (same as the hemisphere).
- Height \( h = 10 \) cm.

\[
V_{\text{cylinder}} = \pi (3)^2 \times 10 = 9\pi \times 10 = 90\pi \text{ cm}^3
\]

### 3. Volume of the Cone
The volume \( V \) of a cone with radius \( r \) and height \( h \) is given by:

\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]

Given:
- Radius \( r = 3 \) cm (same as the hemisphere).
- Height \( h = 8 \) cm.

\[
V_{\text{cone}} = \frac{1}{3} \pi (3)^2 \times 8 = \frac{1}{3} \pi \times 9 \times 8 = 24\pi \text{ cm}^3
\]

### 4. Total Volume of the Toy
The total volume of the toy is the sum of the volumes of the hemisphere, cylinder, and cone:

\[
V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cylinder}} + V_{\text{cone}}
\]

\[
V_{\text{total}} = 18\pi + 90\pi + 24\pi = 132\pi \text{ cm}^3
\]

Thus, the total volume of the toy is:

\[
\boxed{132\pi \text{ cm}^3} \quad \text{or approximately} \quad \boxed{414.69 \text{ cm}^3}
\]

Would you like to explore more about the individual volumes or any other details?

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Here are five related questions you might find interesting:
1. How would the total volume change if the cylinder's height were doubled?
2. What would be the volume if the cone's height was reduced to 4 cm?
3. How do the surface areas of each shape compare in this problem?
4. What would happen to the total volume if the diameter of all shapes was increased by 2 cm?
5. How does the volume of a full sphere compare to the volume of the hemisphere?

**Tip:** When working with composite shapes, break down the problem by analyzing each part separately before summing the results.

A toy consists of a cylinder of diameter 6 cm sandwiched between a hemisphere and a cone of the same diameter. If the cone is of height 8 cm and the cylinder is of height 10 cm, find the total volume of the toy.

Learn how to find the total volume of a composite toy made of a cylinder, hemisphere, and cone. This detailed guide covers volume calculations using geometric formulas for each shape, suitable for students in Grades 8-10.

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