Determine the surface area of eachcomposite object. Give the answers to thenearest tenth.

a) The rectangular prism has dimensions2.5 cm by 2.5 cm by 15.0 cm.The cylinder is 3.5 cm highwith radius 9.6 cm.

To determine the surface area of the composite object, we need to calculate the surface areas of both the rectangular prism and the cylinder individually, and then sum them up. We must also account for any shared surfaces between the two objects, as these surfaces won't contribute to the total external surface area.

Let's break it down:

### 1. **Rectangular Prism:**

Dimensions: \( 2.5 \, \text{cm} \times 2.5 \, \text{cm} \times 15.0 \, \text{cm} \)

The surface area of a rectangular prism is given by:

\[
SA_{\text{prism}} = 2 \left( lw + lh + wh \right)
\]
Where:
- \( l = 15.0 \, \text{cm} \) (length),
- \( w = 2.5 \, \text{cm} \) (width),
- \( h = 2.5 \, \text{cm} \) (height).

Substituting the values:

\[
SA_{\text{prism}} = 2 \left( 15.0 \times 2.5 + 15.0 \times 2.5 + 2.5 \times 2.5 \right)
\]
\[
SA_{\text{prism}} = 2 \left( 37.5 + 37.5 + 6.25 \right) = 2 \times 81.25 = 162.5 \, \text{cm}^2
\]

### 2. **Cylinder:**

Height: \( h = 3.5 \, \text{cm} \), radius: \( r = 9.6 \, \text{cm} \)

The surface area of a cylinder includes the lateral surface area and the area of the two circular bases. The formula for the surface area of a cylinder is:

\[
SA_{\text{cylinder}} = 2\pi r h + 2\pi r^2
\]

Substitute the values for \( r \) and \( h \):

\[
SA_{\text{cylinder}} = 2\pi \times 9.6 \times 3.5 + 2\pi \times (9.6)^2
\]
\[
SA_{\text{cylinder}} = 2\pi \times 33.6 + 2\pi \times 92.16
\]
\[
SA_{\text{cylinder}} = 211.2\pi + 184.32\pi
\]
\[
SA_{\text{cylinder}} = 395.52\pi
\]
\[
SA_{\text{cylinder}} = 1242.8 \, \text{cm}^2 \, (\text{approximately to one decimal place}).
\]

### 3. **Combining Surface Areas:**

The total surface area will be the sum of the surface areas of both objects, minus any surfaces they might share. Since the question does not specify how these objects are connected or if they share a surface, we'll assume they are separate and no shared surface needs to be subtracted.

So, the total surface area is:

\[
SA_{\text{total}} = 162.5 \, \text{cm}^2 + 1242.8 \, \text{cm}^2 = 1405.3 \, \text{cm}^2
\]

### Final Answer:

The total surface area of the composite object is approximately **1405.3 cm²**.

---

Would you like further details on any of the steps? Here are some additional questions to consider:

1. What happens to the surface area if the cylinder is hollow instead of solid?
2. How do the dimensions of a rectangular prism affect its surface area formula?
3. How would the total surface area change if the two objects shared a surface?
4. What would be the surface area of a similar cylinder with a different radius?
5. How would the total surface area change if the prism was a cube?

**Tip:** When solving surface area problems for composite objects, always check if there's any shared surface to avoid over-counting!

Determine the surface area of each composite object. Give the answers to the nearest tenth.

a) The rectangular prism has dimensions 2.5 cm by 2.5 cm by 15.0 cm. The cylinder is 3.5 cm high with radius 9.6 cm.

Learn how to determine the surface area of a composite object consisting of a rectangular prism and a cylinder. This step-by-step solution involves calculating the surface areas of both shapes and summing them up. The rectangular prism has dimensions 2.5 cm x 2.5 cm x 15.0 cm, and the cylinder has a height of 3.5 cm with a radius of 9.6 cm. Suitable for students in Grades 9-11.

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