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:
-
Area of top circle: \[ A_1 = \pi (9^2) = \pi (81) = 81\pi \]
-
Area of bottom circle: \[ A_2 = \pi (3^2) = \pi (9) = 9\pi \]
-
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:
\[ \boxed{270\pi \text{ cm}^2} \quad \text{or approximately } 847.8 \text{ cm}^2. \]