https://file.aiquickdraw.com/mathbot/file/1725982206342r7jyqe7r.jpg

The image shows a cylindrical bottle with a hemispherical top, and it asks to find the total surface area of the bottle. The measurements provided are:

- Height of the cylindrical part = 9 cm
- Radius of the base = 3.5 cm

To calculate the total surface area, we need to find the areas of both the cylindrical part and the hemispherical part. Here's the breakdown:

### 1. **Surface Area of the Cylinder** (without top and bottom):
The lateral surface area (side surface) of a cylinder is given by:
\[
A_{\text{cylinder}} = 2\pi r h
\]
where:
- \( r = 3.5 \) cm (radius),
- \( h = 9 \) cm (height).

### 2. **Surface Area of the Hemisphere**:
The surface area of a hemisphere (including the base) is given by:
\[
A_{\text{hemisphere}} = 2\pi r^2
\]
where \( r = 3.5 \) cm.

### 3. **Total Surface Area**:
Since the hemisphere covers the top of the cylinder, we don't count the base of the cylinder. Therefore, the total surface area is:
\[
A_{\text{total}} = 2\pi r h + 2\pi r^2
\]

Let me calculate this for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you would like to calculate it manually, here's the process:

1. **Lateral Surface Area of the Cylinder**:
   \[
   A_{\text{cylinder}} = 2\pi \times 3.5 \times 9
   \]

2. **Surface Area of the Hemisphere**:
   \[
   A_{\text{hemisphere}} = 2\pi \times (3.5)^2
   \]

3. **Total Surface Area**:
   Add the two areas together.

Would you like to continue solving it by hand, or try the calculation again later?

Here are some related questions you might find helpful:
1. How do you calculate the surface area of a cone?
2. What is the formula for the volume of a hemisphere?
3. How does the formula for a full sphere's surface area differ from that of a hemisphere?
4. How does the surface area change if the cylinder has a different height?
5. How can we use the same concept to calculate the surface area of other 3D objects?

**Tip**: Always check the units of measurement when calculating surface areas or volumes. Consistency in units helps avoid errors.

Find the total surface area of the bottle.

Learn how to calculate the total surface area of a bottle with a cylindrical body and a hemispherical top. The problem involves finding the surface area of the cylinder and the hemisphere using geometric formulas. Suitable for students in Grades 8-10.

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