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To solve for the **lateral surface area** and **total surface area** of the cone, let's go through each part step-by-step.

### Given:
- Base radius (\( r \)) = 3 in
- Slant height (\( s \)) = 12 in
- Use \( \pi = 3.14 \)

### Formulas:
1. **Lateral Surface Area (LSA) of a cone**:
   \[
   \text{LSA} = \pi \cdot r \cdot s
   \]
2. **Total Surface Area (SA) of a cone** (which includes the base):
   \[
   \text{SA} = \pi \cdot r \cdot s + \pi \cdot r^2
   \]

### Step 1: Calculate the Lateral Surface Area
Using the values:
\[
\text{LSA} = 3.14 \cdot 3 \cdot 12 = 3.14 \cdot 36 = 113.04 \text{ in}^2
\]

### Step 2: Calculate the Total Surface Area
The area of the base is given by \( \pi \cdot r^2 \).
\[
\text{Base Area} = 3.14 \cdot 3^2 = 3.14 \cdot 9 = 28.26 \text{ in}^2
\]
Adding this to the LSA:
\[
\text{SA} = 113.04 + 28.26 = 141.3 \text{ in}^2
\]

### Answers:
- **(a) Lateral Surface Area**: \( 113.04 \, \text{in}^2 \)
- **(b) Surface Area**: \( 141.3 \, \text{in}^2 \)

Would you like more details on any step, or have any questions?

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Here are 5 related questions for further practice:
1. How would you calculate the volume of the cone with the same dimensions?
2. What changes in the formulas if the cone’s height (not slant height) is given instead?
3. If the slant height is doubled, how does that affect the lateral surface area?
4. How would the surface area change if the radius of the cone is increased by 1 inch?
5. Why is it important to distinguish between slant height and vertical height in cone calculations?

**Tip**: For cones, always confirm whether you're given the slant height or the actual height, as each has different applications in surface area and volume calculations.

Find the lateral surface area and surface area of a cone with a base radius of 3 in and a slant height of 12 in. Use the value 3.14 for π, and do not do any rounding.

Learn how to find the lateral surface area and total surface area of a cone with a radius of 3 inches and slant height of 12 inches. This problem covers geometry concepts, including the lateral surface area formula and the total surface area formula for cones.

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