https://file.aiquickdraw.com/mathbot/file/1726716358397fydehxin.jpg

To determine the surface area of the composite object shown in the image, we need to break the object down into simpler geometric shapes. From the image, we can identify two main parts:

1. **A cylindrical base**: 
   - Height: 4.5 cm
   - Radius: 1.0 cm

2. **A disk (cylinder with very short height)**: 
   - Height (thickness): 1.5 cm
   - Radius: 4.0 cm

### Step 1: Surface Area of the Cylindrical Base

The formula for the surface area of a cylinder is given by:
\[
A_{\text{cylinder}} = 2\pi r h + 2\pi r^2
\]
Where:
- \( r = 1.0 \, \text{cm} \) (radius)
- \( h = 4.5 \, \text{cm} \) (height)

The surface area consists of:
- The **lateral area**: \( 2\pi rh \)
- The **two circular ends**: \( 2\pi r^2 \)

### Step 2: Surface Area of the Disk

The disk can be considered as a short cylinder. Its surface area is calculated similarly:
\[
A_{\text{disk}} = 2\pi r_{\text{disk}} h_{\text{disk}} + 2\pi r_{\text{disk}}^2
\]
Where:
- \( r_{\text{disk}} = 4.0 \, \text{cm} \)
- \( h_{\text{disk}} = 1.5 \, \text{cm} \)

### Step 3: Adjustments for Overlapping Surfaces

Since the disk is attached to the cylindrical base, one of the circular ends of the cylindrical base is not exposed. So, we subtract the area of that circular end to avoid counting it twice.

Now, let's calculate the total surface area based on these steps.

I'll compute it for you.The total surface area of the composite object is approximately **169.6 cm²**, rounded to the nearest tenth.

Do you have any questions or would you like further details?

Here are five related questions to deepen your understanding:

1. How would the total surface area change if the height of the disk were doubled?
2. How is surface area affected if we increase only the radius of the cylindrical base?
3. Can you calculate the volume of this composite object?
4. What happens to the surface area if we remove the cylindrical base completely?
5. How do you calculate surface area for other composite shapes, such as a cone or sphere attached to a cylinder?

**Tip**: When calculating surface areas of composite shapes, always remember to account for any overlapping or hidden surfaces.

Determine the surface area of the composite object, consisting of a cylindrical base and a disk. Dimensions include: base radius = 1.0 cm, base height = 4.5 cm, disk radius = 4.0 cm, disk height = 1.5 cm.

This guide provides a detailed solution to calculating the surface area of a composite object made of a cylinder and a disk. Learn how to apply formulas for cylindrical and disk-shaped objects and adjust for overlapping surfaces. Dimensions include a 1.0 cm radius and 4.5 cm height for the cylindrical base, and a 4.0 cm radius and 1.5 cm height for the disk.

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