https://file.aiquickdraw.com/mathbot/file/17298552975362fx6zvmu.jpg

The object in the image is a cone and cylinder combined into one shape. To calculate the total surface area, we need to consider the surface areas of both the cone and the cylinder, excluding the circular base that they share.

### Step 1: Identify the given dimensions
- **Radius** of the base (common to both the cylinder and cone): \( r = 2 \, \text{cm} \)
- **Height of the cylinder**: \( h_{\text{cyl}} = 3 \, \text{cm} \)
- **Slant height of the cone**: \( l_{\text{cone}} = 2.5 \, \text{cm} \)

### Step 2: Surface area formulas
- **Surface area of the cylinder** (excluding the base, since it's shared with the cone):
  \[
  A_{\text{cyl}} = 2\pi r h_{\text{cyl}}
  \]
- **Lateral surface area of the cone**:
  \[
  A_{\text{cone}} = \pi r l_{\text{cone}}
  \]

### Step 3: Calculate individual surface areas

- **Cylinder lateral surface area**:
  \[
  A_{\text{cyl}} = 2\pi (2) (3) = 12\pi \, \text{cm}^2
  \]

- **Cone lateral surface area**:
  \[
  A_{\text{cone}} = \pi (2) (2.5) = 5\pi \, \text{cm}^2
  \]

### Step 4: Total surface area
Since there is no need to include the circular base (as it is part of both the cone and cylinder), we sum the lateral surface areas:

\[
A_{\text{total}} = A_{\text{cyl}} + A_{\text{cone}} = 12\pi + 5\pi = 17\pi \, \text{cm}^2
\]

### Final Answer:
\[
A_{\text{total}} = 17\pi \approx 53.41 \, \text{cm}^2
\]

Let me know if you'd like further clarification or details!

### Related questions:
1. How would the surface area change if the height of the cylinder were doubled?
2. What is the surface area if the radius of the base is increased to 3 cm?
3. How would you calculate the volume of the cone and cylinder combined?
4. What if the object had a circular base at both ends?
5. How does the slant height of the cone affect its surface area?

### Tip:
Always distinguish between lateral surface area and total surface area when dealing with composite shapes.

Calculate the total surface areas of the objects shown.

Learn how to calculate the surface area of a composite shape consisting of a cylinder and cone. Given a radius of 2 cm, a cylinder height of 3 cm, and a cone slant height of 2.5 cm, follow the step-by-step breakdown of surface area calculations using key geometric formulas.

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