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?

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.