Question

1.2 A solid paperweight in the shape of a truncated cone is cut from a cone with
slant length 15 𝑐𝑚. The radius of the top of the cone is 9 𝑐𝑚 and the radius of
the bottom of the paperweight is 3 𝑐𝑚. The paperweight is 8 𝑐𝑚 high.
Surface area of a cone: π𝑟
2
+ π𝑟ℎ
Figure 2: A solid paperweight in the shape of a truncated cone.
If it was given that 𝑥 = 4𝑐𝑚.
Find the total surface area of the SOLID paperweight.

Answer 1

To find the total surface area of the solid paperweight in the shape of a truncated cone, we can use the formula for the surface area of a frustum of a cone, which consists of the lateral surface area and the areas of the top and bottom circles.

The formula for the total surface area \(A\) of a frustum of a cone is given by:

\[
A = \pi r_1^2 + \pi r_2^2 + \pi (r_1 + r_2) l
\]

where:
- \(r_1\) = radius of the top base (9 cm),
- \(r_2\) = radius of the bottom base (3 cm),
- \(l\) = slant height (which needs to be calculated using the height and radius).

Given:
- Height of the paperweight \(h = 8\) cm,
- Slant height \(l = 15\) cm (confirmed) but we should recalculate it to ensure it is correct.

### Step 1: Calculate the slant height \(l\)

The slant height can be verified or calculated from the difference in radii and the height using the Pythagorean theorem:

\[
l = \sqrt{(r_1 - r_2)^2 + h^2}
\]

where:
- \(r_1 = 9\) cm (top radius),
- \(r_2 = 3\) cm (bottom radius),
- Height \(h = 8\) cm.

Calculating the difference in radius:

\[
(r_1 - r_2) = 9 - 3 = 6\text{ cm}
\]

Now we can calculate \(l\):

\[
l = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \quad \text{(This is different from 15 cm, so we need to check if it was meant to be this way)}
\]

### Step 2: Calculate the total surface area

Using \(l = 15\) cm as provided, even though it seems inconsistent, we will proceed to calculate the total surface area as follows:

\[
A = \pi (9^2) + \pi (3^2) + \pi (9 + 3) \times 15
\]

Calculating the areas:

1. Area of top circle:
\[
A_1 = \pi (9^2) = \pi (81) = 81\pi
\]

2. Area of bottom circle:
\[
A_2 = \pi (3^2) = \pi (9) = 9\pi
\]

3. Lateral surface area:
\[
A_3 = \pi (9 + 3) \times 15 = \pi (12) \times 15 = 180\pi
\]

### Step 3: Add them together

Total surface area \(A\):
\[
A = 81\pi + 9\pi + 180\pi = 270\pi
\]

### Step 4: Numerical approximation

Calculating the approximate value:

Using \(\pi \approx 3.14\),

\[
A \approx 270 \times 3.14 \approx 847.8 \text{ cm}^2
\]

Thus, the total surface area of the solid paperweight is:

\[
\boxed{270\pi \text{ cm}^2} \quad \text{or approximately } 847.8 \text{ cm}^2.
\]