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We need to calculate the exterior surface area of the open half-cylinder shown in the image.

The half-cylinder has three components contributing to the surface area:
1. The curved surface of the half-cylinder.
2. The flat rectangular surface at the bottom.
3. One flat semicircular face at the end (note: the other end is open).

### Step 1: **Curved Surface Area**
The formula for the surface area of a full cylinder's side is:
\[
\text{Curved Surface Area (full cylinder)} = 2\pi r h
\]
For a half-cylinder, we only need half of this surface area:
\[
\text{Curved Surface Area (half-cylinder)} = \pi r h
\]
Where:
- \( r \) is the radius of the cylinder (half of 8.4 cm, since it's the diameter).
- \( h \) is the height of the cylinder, which is given as 9.6 cm.

So:
\[
r = \frac{8.4}{2} = 4.2 \, \text{cm}
\]
\[
\text{Curved Surface Area} = \pi \times 4.2 \times 9.6
\]

### Step 2: **Flat Rectangular Surface (Bottom)**
The rectangular bottom surface is simply the length \( h \) multiplied by the diameter \( d \) (8.4 cm):
\[
\text{Bottom Surface Area} = h \times d = 9.6 \times 8.4
\]

### Step 3: **Semicircular Face**
The area of a full circle is \( \pi r^2 \), but since we only have a semicircle, the formula becomes:
\[
\text{Semicircular Face Area} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4.2)^2
\]

Now, let's calculate all these components and sum them up to get the total surface area.The total exterior surface area of the open half-cylinder is \( 235.0 \, \text{cm}^2 \) (rounded to 1 decimal place).

Would you like any further explanation or details on the steps? Here are some related questions to deepen your understanding:

1. How would the surface area change if the half-cylinder were fully closed?
2. What would the surface area be if only the curved surface was required?
3. How do you calculate the surface area for a full cylinder instead of a half-cylinder?
4. What is the significance of rounding in such calculations, and how might it affect precision?
5. How could we extend this to calculate the volume of the half-cylinder?

**Tip:** Always ensure to identify which surfaces contribute to the total area in composite shapes like this.

Find the exterior surface area of this open half-cylinder with a radius of 4.2 cm (half of 8.4 cm), and height of 9.6 cm.

Learn how to find the exterior surface area of an open half-cylinder. The problem involves key geometry concepts like surface area, and formulas such as the curved surface area and semicircular face area. This problem is suitable for students in Grades 9-11, providing a step-by-step solution.

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