Find the total volume of composite rectangular prisms in this math worksheet.

Let's solve each composite rectangular prism problem step by step. We'll break each figure into simpler rectangular prisms, calculate their individual volumes, and then add them together.

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Formula for Volume of a Rectangular Prism:

$$
V = \text{length} \times \text{width} \times \text{height}
$$

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a.

Figure Description:
- The shape is made of two stacked rectangles.
- Bottom part:
- Length = 14 cm
- Width = 3 cm (given)
- Height = 5 cm
- Top part:
- Same length and width as bottom: 14 cm × 3 cm
- Height = 5 cm

So both parts are identical.

Volume of one block:
$$
V = 14 \times 3 \times 5 = 210\ \text{cm}^3
$$

Total volume:
$$
V_{\text{total}} = 210 + 210 = \boxed{420}\ \text{cm}^3
$$

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b.

Figure Description:
- Two rectangular prisms side by side.
- Left block:
- Length = 15 in (but only the left portion is 6 in high)
- Width = 4 in
- Height = 6 in
- But wait — the total base is 15 in, and the right block is 7 in long, so the left block must be:
- Length = 15 - 7 = 8 in? Wait, let’s look carefully.

Actually, the figure shows:
- A larger base of 15 in wide.
- The lower part has height 4 in, and extends the full 15 in.
- On top of that, a smaller box of 7 in long, 3 in high, placed on the right side.

Wait — actually, it looks like:
- The bottom layer is a rectangle of:
- Length = 15 in
- Width = 4 in
- Height = 4 in → So this is the base.

Then on top of that, a smaller box:
- Length = 7 in
- Width = ? Wait — we don’t have width given. But from the image, likely the depth (width) is consistent.

But the figure shows:
- The entire depth is not labeled, but since it's a 3D figure, we assume the depth (width) is uniform.

Looking at the dimensions:
- The bottom base is 15 in long, 4 in high, and we assume depth is same as the top block.
- The top block is 7 in long, 3 in high, and sits on the right side.

But what is the depth (width)?

Wait — the vertical dimension is 6 in on the left side, but the bottom is 4 in high, and the top is 3 in high. So the total height is 6 in.

But the bottom is 4 in high, and the top is 3 in high, and they are stacked vertically?

No — looking again:

The figure shows:
- A large base of 15 in long, 4 in high, and depth (width) not labeled.
- Then a smaller block of 7 in long, 3 in high, sitting on top of the right end of the base.

But the height of the base is 4 in, and the top block is 3 in high, so total height is 4 + 3 = 7 in? But the label says 6 in.

Wait — the label says:
- On the left, the total height is 6 in.
- The bottom is 4 in high.
- So the top block must be 6 - 4 = 2 in high? But it says 3 in.

Hmm — contradiction?

Wait — let’s re-express:

From the diagram:
- The bottom part is 4 in high.
- The top part is 3 in high.
- The total height on the left is 4 in, and on the right, it’s 4 + 3 = 7 in? But it says 6 in.

Wait — no. The label on the left says 6 in — that’s the total height of the entire structure?

But the bottom is 4 in, and the top is 3 in — that would make 7 in.

Wait — perhaps the bottom is 6 in high, and the top is 3 in high?

But the label says "6 in" next to the bottom, and "3 in" on top.

Wait — let's read carefully:

> b.
> Diagram:
> - A base with height 6 in, width 4 in, length 15 in?
> - Then a smaller box on top, length 7 in, height 3 in.

But if the base is 6 in high, and the top is 3 in high, then the total height is 9 in — but it doesn't show that.

Wait — no: the base is 6 in high? But the label says “6 in” next to the vertical side, and “4 in” below it?

Wait — I think there's confusion in interpretation.

Let me interpret based on standard labeling.

Looking at the image description:

- The bottom rectangular prism:
- Length = 15 in
- Width = 4 in (depth)
- Height = 6 in? But wait — the label says "6 in" on the left side, and "4 in" below it.

Ah! It appears:
- The height of the bottom part is 4 in.
- The height of the top part is 3 in.
- The total height on the right is 4 + 3 = 7 in, but the left side shows 6 in?

That can't be.

Wait — maybe the bottom is 6 in high, and the top is 3 in high, and the total height is 6 in?

But the label says “6 in” on the left, and “4 in” below it.

Wait — perhaps the depth is 4 in? That makes more sense.

Let’s try this:

- The bottom prism:
- Length = 15 in
- Depth (width) = 4 in
- Height = 6 in? But then the top block is 3 in high.

But the top block is placed on the right, and its height is labeled 3 in.

But the total height on the left is 6 in, and on the right is 6 + 3 = 9 in? But that doesn’t match.

Wait — perhaps the bottom is 6 in high, and the top is 3 in high, and the top is offset.

But the label says:
- Vertical line on the left: 6 in
- Horizontal line on the bottom: 4 in — that’s likely the depth or width.
- Then the top block is 3 in high, and 7 in long.

But the bottom is 15 in long.

So the bottom prism:
- Length = 15 in
- Width = 4 in (depth)
- Height = 6 in

The top prism:
- Length = 7 in
- Width = 4 in (same depth)
- Height = 3 in

But the top block is placed on top of the right end of the bottom block.

So volume:

- Bottom: $ V_1 = 15 \times 4 \times 6 = 360\ \text{in}^3 $
- Top: $ V_2 = 7 \times 4 \times 3 = 84\ \text{in}^3 $

Total volume:
$$
V = 360 + 84 = \boxed{444}\ \text{in}^3
$$

But wait — the height of the bottom is 6 in, and the top is 3 in — so total height is 9 in, but the label shows 6 in on the left? That suggests the bottom is only 6 in high, and the top is not adding height on the left.

But the top is only on the right.

So yes, the bottom is 6 in high, and the top adds 3 in on the right.

So the total height on the left is 6 in, on the right is 9 in.

That matches.

So answer: $ \boxed{444} $ in³

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c.

Figure Description:
- L-shaped composite prism.
- Can be split into two rectangles.

We can split it into:
- Bottom rectangle: 10 cm long, 3 cm wide, 2 cm high
- Top rectangle: 6 cm long, 4 cm high, but width?

Wait — the depth (width) is not labeled, but from the diagram, likely the depth is the same for both.

Looking:
- The bottom part is 10 cm long, 3 cm high, and depth?
- The top part is 6 cm long, 4 cm high, and sits on the right end of the bottom.

But the depth (width) is missing.

Wait — the bottom has height 2 cm, and the top has height 4 cm, and the total height is 6 cm?

But the bottom is 3 cm high? No.

Wait — labels:
- Bottom part: height = 2 cm
- Top part: height = 4 cm
- Total height = 6 cm
- Bottom length = 10 cm
- Top length = 6 cm
- Depth (width) — not labeled.

But the bottom is 10 cm long, 2 cm high, and depth?
- The top is 6 cm long, 4 cm high, and same depth?

But the top is sitting on the right end of the bottom.

But the depth is not given.

Wait — perhaps the depth is implied to be the same throughout.

But it's not labeled.

Wait — look: the bottom has a horizontal line labeled 3 cm — is that the depth?

Yes — likely, the depth (width) is 3 cm.

So:
- Bottom rectangle:
- Length = 10 cm
- Width = 3 cm
- Height = 2 cm
- $ V_1 = 10 \times 3 \times 2 = 60\ \text{cm}^3 $

- Top rectangle:
- Length = 6 cm
- Width = 3 cm (same depth)
- Height = 4 cm
- $ V_2 = 6 \times 3 \times 4 = 72\ \text{cm}^3 $

Total volume:
$$
V = 60 + 72 = \boxed{132}\ \text{cm}^3
$$

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d.

Figure Description:
- Two blocks stacked.
- Bottom block:
- Length = 10 m
- Width = 3 m
- Height = 6 m
- Top block:
- Length = 8 m
- Width = ? Wait — the depth is not labeled.

But the total height is 12 m, and the bottom is 6 m high, so the top is 12 - 6 = 6 m high.

Also, the top block is 8 m long, and sits on top of the bottom.

But the depth (width) — is it the same?

Looking at the figure:
- The bottom has width 3 m (labeled).
- The top block is wider? No — the depth (width) is not labeled.

Wait — the side view shows:
- Bottom block: 6 m high, 3 m deep
- Top block: 6 m high, 8 m long, and depth?

But the top block is centered on the bottom?

But the bottom is 10 m long, and the top is 8 m long — so it fits.

But what is the depth (width) of the top block?

It's not labeled. But likely, the depth is the same as the bottom, which is 3 m.

Wait — the top block is shown to be wider than the bottom?

No — the side shows the depth is 3 m for both.

But the top block is shorter in length (8 m vs 10 m), but same depth.

So:
- Bottom block:
- $ V_1 = 10 \times 3 \times 6 = 180\ \text{m}^3 $
- Top block:
- $ V_2 = 8 \times 3 \times 6 = 144\ \text{m}^3 $

Total volume:
$$
V = 180 + 144 = \boxed{324}\ \text{m}^3
$$

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✔ Final Answers:

a. $ \boxed{420}\ \text{cm}^3 $
b. $ \boxed{444}\ \text{in}^3 $
c. $ \boxed{132}\ \text{cm}^3 $
d. $ \boxed{324}\ \text{m}^3 $

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Let me know if you'd like diagrams or further explanation!