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The problem you've provided involves finding the total surface area of a hollow cone with given inner and outer dimensions.

### Given Data:
- **Inner radius of cone (r₁)** = 5
- **Outer radius of cone (r₂)** = 8
- **Inner height of cone (h₁)** = 12
- **Total height of cone (h₂)** = 15

To find the total surface area of the hollow cone, we must consider two components:
1. **The lateral surface area** of both the inner and outer cones.
2. **The base area**, which is the ring formed by the difference between the outer and inner radii.

### Step 1: Lateral Surface Area of the Outer and Inner Cones
The formula for the lateral surface area (A) of a cone is:

\[
A = \pi r \cdot l
\]

where \( r \) is the radius and \( l \) is the slant height.

#### Slant height (l)
The slant height \( l \) can be calculated using the Pythagorean theorem:

\[
l = \sqrt{r^2 + h^2}
\]

#### Slant height for outer cone:
For the outer cone (radius = 8, height = 15):

\[
l_2 = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17
\]

#### Slant height for inner cone:
For the inner cone (radius = 5, height = 12):

\[
l_1 = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
\]

### Step 2: Total Lateral Surface Area
Now, calculate the lateral surface area for both the inner and outer cones:

- **Outer cone lateral surface area**:

\[
A_2 = \pi \cdot 8 \cdot 17 = 136\pi
\]

- **Inner cone lateral surface area**:

\[
A_1 = \pi \cdot 5 \cdot 13 = 65\pi
\]

The lateral surface area of the hollow cone is the difference between the outer and inner lateral areas:

\[
A_{lateral} = A_2 - A_1 = 136\pi - 65\pi = 71\pi
\]

### Step 3: Base Area
The base area is the area of the outer circle minus the area of the inner circle. The formula for the area of a circle is \( \pi r^2 \).

\[
A_{base} = \pi \cdot (r_2^2 - r_1^2) = \pi \cdot (8^2 - 5^2) = \pi \cdot (64 - 25) = \pi \cdot 39 = 39\pi
\]

### Step 4: Total Surface Area
The total surface area is the sum of the lateral surface area and the base area:

\[
A_{total} = A_{lateral} + A_{base} = 71\pi + 39\pi = 110\pi
\]

However, based on the data, something seems off because none of the options directly match 110π. Let's recheck if we need to adjust the interpretation or options, but it seems clear the calculated total surface area is \( 110\pi \).

Would you like any further clarifications?

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Here are 5 questions that follow this explanation:
1. How does the slant height affect the surface area of the cone?
2. Can the surface area be calculated without the slant height?
3. What would happen if the cone had no inner cone, just a solid cone?
4. How is the surface area formula derived for a cone?
5. How can this concept be extended to other shapes, like frustums?

Tip: For cones, always verify both slant height and radius before computing surface area.

Find the total surface area of a hollow cone with inner radius = 5, outer radius = 8, inner height = 12, and total height = 15. Given options: a) 201 pi, b) 215 pi, c) 240 pi, d) 280 pi.

Learn how to find the total surface area of a hollow cone with given dimensions: inner radius = 5, outer radius = 8, inner height = 12, and total height = 15. Includes step-by-step calculation of the lateral and base areas using key geometry formulas.

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