https://file.aiquickdraw.com/mathbot/file/1726332830173aa7qeqfl.jpg

Let's analyze the image and solve each part of the problem step by step.

The solid in the diagram consists of two rectangular blocks, with measurements given in centimeters (cm):

- The larger block has dimensions: 5 cm (length), 4 cm (width), and 2 cm (height).
- The smaller block on top has dimensions: 3 cm (length), 2 cm (width), and 2 cm (height).

### a) Surface Area of the Solid

We calculate the surface area by finding the area of all visible faces of the solid and adding them together. The solid has multiple faces, and some faces are hidden or overlap. We'll need to subtract overlapping areas.

1. **Bottom block:**
   - Two sides (front and back): \(5 \, \text{cm} \times 2 \, \text{cm} = 10 \, \text{cm}^2\) (×2 for both sides = \(20 \, \text{cm}^2\)).
   - Two sides (left and right): \(4 \, \text{cm} \times 2 \, \text{cm} = 8 \, \text{cm}^2\) (×2 for both sides = \(16 \, \text{cm}^2\)).
   - Top face (visible part of the bottom block): Since part of the top block covers some area, the visible part is: \(2 \, \text{cm} \times 4 \, \text{cm} = 8 \, \text{cm}^2\).
   - Bottom face: \(5 \, \text{cm} \times 4 \, \text{cm} = 20 \, \text{cm}^2\).

2. **Top block:**
   - Two sides (front and back): \(3 \, \text{cm} \times 2 \, \text{cm} = 6 \, \text{cm}^2\) (×2 for both sides = \(12 \, \text{cm}^2\)).
   - Two sides (left and right): \(2 \, \text{cm} \times 2 \, \text{cm} = 4 \, \text{cm}^2\) (×2 for both sides = \(8 \, \text{cm}^2\)).
   - Top face: \(3 \, \text{cm} \times 2 \, \text{cm} = 6 \, \text{cm}^2\).
   - The bottom of the top block overlaps the top of the bottom block, so we do not count this area again.

Now, adding all these areas together:

\[
\text{Surface area} = 20 + 16 + 8 + 20 + 12 + 8 + 6 = 90 \, \text{cm}^2
\]

Thus, the surface area of the solid is **90 cm²**.

---

### b) Area of the Cross-Section of the Solid

A cross-section typically refers to a slice through the object. If we take a vertical cross-section along the width (through the height and depth), we get a shape consisting of two rectangles.

- The bottom rectangle has dimensions: 2 cm (height) × 4 cm (width) = 8 cm².
- The top rectangle has dimensions: 2 cm (height) × 2 cm (width) = 4 cm².

So, the total cross-sectional area is:

\[
\text{Cross-sectional area} = 8 \, \text{cm}^2 + 4 \, \text{cm}^2 = 12 \, \text{cm}^2
\]

Thus, the cross-sectional area is **12 cm²**.

---

### c) Volume of Metal Needed to Make this Shape

The volume of the solid can be found by summing the volumes of the two blocks.

1. **Bottom block:**
   \[
   \text{Volume of bottom block} = 5 \, \text{cm} \times 4 \, \text{cm} \times 2 \, \text{cm} = 40 \, \text{cm}^3
   \]

2. **Top block:**
   \[
   \text{Volume of top block} = 3 \, \text{cm} \times 2 \, \text{cm} \times 2 \, \text{cm} = 12 \, \text{cm}^3
   \]

So, the total volume is:

\[
\text{Total volume} = 40 \, \text{cm}^3 + 12 \, \text{cm}^3 = 52 \, \text{cm}^3
   \]

Thus, the volume of metal needed to make the shape is **52 cm³**.

---

Would you like more details on any step? Here are some related questions you might explore:

1. How do you calculate the volume of composite solids in general?
2. How would the surface area change if the top block had a different height?
3. Can you calculate the lateral surface area of just one of the blocks?
4. What happens to the cross-sectional area if you cut the solid in a different plane?
5. How do you calculate the surface area of more complex solids?

**Tip:** When calculating surface area for composite solids, be sure to subtract overlapping areas to avoid double counting.

Use the measurements on the diagram to find a) the surface area of the solid, b) the area of cross-section of the solid, c) the volume of metal needed to make this shape.

Learn how to calculate the surface area, cross-sectional area, and volume of a composite rectangular solid using geometry. This problem covers the key concepts of surface area and volume for rectangular prisms, suitable for students in Grades 6-9.

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