Volume of Composite Solids worksheet featuring nine geometric figures with dimensions for calculating volume.

Let's solve three of the problems from the worksheet, making sure to include question #5, as required.

We’ll solve:

- Problem 1
- Problem 5
- Problem 9

---

🔷 Problem 1: Volume of an L-shaped prism (in³)

This solid is made of two rectangular prisms joined together.

#### Step 1: Break into two rectangles

We can split this L-shape into:
- A vertical rectangular prism on the left
- A horizontal rectangular prism on the right

But let’s analyze dimensions carefully:

From the image:
- The entire base is 12 in long.
- The right section is 8 in wide and 3 in high.
- The left section extends up to 10 in high, and its width is $12 - 8 = 4$ in.

So:
- Left part:
Length = 4 in (width), Height = 10 in, Depth = 3 in
Volume = $4 \times 10 \times 3 = 120\ \text{in}^3$

- Right part:
Length = 8 in, Height = 3 in, Depth = 3 in
Volume = $8 \times 3 \times 3 = 72\ \text{in}^3$

#### ✔ Total Volume:
$$
120 + 72 = \boxed{192\ \text{in}^3}
$$

---

🔷 Problem 5: Composite solid with a cylinder on top of a cube (in³)

This shape has:
- A cube at the bottom (5 in × 5 in × 5 in)
- A cylinder on top (diameter = 4 in, height = 4 in)

Use $ \pi = 3.14 $

#### Step 1: Volume of cube
$$
V_{\text{cube}} = 5 \times 5 \times 5 = 125\ \text{in}^3
$$

#### Step 2: Volume of cylinder
$$
V_{\text{cylinder}} = \pi r^2 h
$$
- Radius $r = \frac{4}{2} = 2$ in
- Height $h = 4$ in

$$
V = 3.14 \times (2)^2 \times 4 = 3.14 \times 4 \times 4 = 3.14 \times 16 = 50.24\ \text{in}^3
$$

#### ✔ Total Volume:
$$
125 + 50.24 = \boxed{175.24\ \text{in}^3}
$$

---

🔷 Problem 9: T-shaped block (in³)

This is a composite solid made of three rectangular prisms (or two if we group smartly).

Dimensions:
- Base: 4 in (length), 2 in (height), 1 in (depth)
- Middle vertical piece: 1 in (length), 3 in (height), 1 in (depth)
- But wait — looking closely: it’s a T-shape with:
- Horizontal base: 4 in long, 1 in high, 1 in deep
- Vertical stem: 1 in wide, 3 in high, 1 in deep

But the vertical part is centered, so it's 1 in wide, 3 in tall, and 1 in deep.

However, the horizontal base is 4 in long, 1 in high, 1 in deep.

But the vertical part sits on top of the horizontal base, overlapping by 1 in.

So total volume = volume of horizontal base + volume of vertical stem.

#### Step 1: Volume of horizontal base
$$
V = 4 \times 1 \times 1 = 4\ \text{in}^3
$$

#### Step 2: Volume of vertical stem
$$
V = 1 \times 3 \times 1 = 3\ \text{in}^3
$$

Note: There is no overlap in volume since both are solid and stacked. We just add.

#### ✔ Total Volume:
$$
4 + 3 = \boxed{7\ \text{in}^3}
$$

---

✔ Final Answers:

| Problem | Volume |
|--------|--------|
| 1 | 192 in³ |
| 5 | 175.24 in³ |
| 9 | 7 in³ |

> ✔ All answers are rounded appropriately (problem 5 uses π ≈ 3.14, so 50.24 is correct).

Let me know if you'd like help with any other problems!