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Calculate the surface area of the closed solid in Figure 3. The height from the top of the half sphere to the bottom of the right cone is 16 𝑐𝑚 and the radius of the sphere is 4 𝑐𝑚. Leave your answer in terms of π.

To calculate the surface area of the closed solid in Figure 3, which consists of a cone and half of a sphere, we will proceed step-by-step using the given formulas and measurements.

### Given:
- The **radius** \( r \) of the sphere (and the base of the cone) = 4 cm.
- The **height** from the top of the half-sphere to the bottom of the cone = 16 cm.
- This height includes the radius of the half-sphere, so the height of the cone itself \( h \) is:
  \[
  h_{\text{cone}} = 16 \, \text{cm} - 4 \, \text{cm} = 12 \, \text{cm}.
  \]

Now we calculate the surface area in two parts:

### 1. **Surface area of the half-sphere:**
The formula for the surface area of a **full** sphere is:
\[
A_{\text{sphere}} = 4\pi r^2
\]
Since we have **half** of a sphere, the surface area is:
\[
A_{\text{half-sphere}} = \frac{1}{2} \times 4\pi r^2 = 2\pi r^2
\]
Substituting \( r = 4 \, \text{cm} \):
\[
A_{\text{half-sphere}} = 2\pi (4)^2 = 2\pi \times 16 = 32\pi \, \text{cm}^2
\]

### 2. **Surface area of the cone:**
The formula for the surface area of a right cone (excluding the base) is:
\[
A_{\text{cone}} = \pi r s
\]
where \( s \) is the slant height of the cone, which can be found using the Pythagorean theorem:
\[
s = \sqrt{r^2 + h^2}
\]
Substituting \( r = 4 \, \text{cm} \) and \( h = 12 \, \text{cm} \):
\[
s = \sqrt{4^2 + 12^2} = \sqrt{16 + 144} = \sqrt{160} = 4\sqrt{10} \, \text{cm}
\]
Now, substituting this into the surface area formula for the cone:
\[
A_{\text{cone}} = \pi \times 4 \times 4\sqrt{10} = 16\pi\sqrt{10} \, \text{cm}^2
\]

### 3. **Total surface area:**
Since the base of the cone and the flat part of the half-sphere are not exposed, we do not include the area of the base. Thus, the total surface area is the sum of the surface area of the half-sphere and the lateral surface area of the cone:
\[
A_{\text{total}} = A_{\text{half-sphere}} + A_{\text{cone}} = 32\pi + 16\pi\sqrt{10} \, \text{cm}^2
\]

Thus, the surface area of the closed solid is:
\[
\boxed{32\pi + 16\pi\sqrt{10} \, \text{cm}^2}
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you calculate the volume of the cone and half-sphere in this shape?
2. What is the formula for the volume of a cone?
3. How do you calculate the slant height of a cone in different situations?
4. How does surface area change if the radius of the base increases?
5. What is the difference between lateral surface area and total surface area?

**Tip:** When dealing with composite shapes, breaking the problem into smaller parts (like cone and sphere) helps simplify complex surface area calculations.

Learn how to calculate the surface area of a closed solid, consisting of a cone and half of a sphere, using geometric formulas. The radius is 4 cm, and the total height is 16 cm. Detailed step-by-step solution provided, suitable for high school students.

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