Question

the solid below is made from cubes. find its volume. (there is a diagram of a rectangular prism made of small cubes with 1 in indicated for the edge length of a small cube)

Explanation:

Step1: Determine the dimensions

From the figure, we can see that the length, width, and height of the rectangular prism (made of cubes) are 4 units, 3 units, and 4 units respectively? Wait, no, looking at the figure, the number of cubes along the length: let's count. Wait, the base layer: how many cubes? Wait, the side with 1 in: each cube has an edge length of 1 inch. So let's count the number of cubes along each dimension.

Looking at the figure, along the length (let's say x - axis): 4 cubes? Wait, no, wait the front face: the number of cubes along the width (y - axis) is 3, along the height (z - axis) is 4, and along the length (x - axis) is 4? Wait, no, maybe I miscounted. Wait, the figure: let's see, the base (top layer) has a grid. Let's count the number of cubes along each side.

Wait, the front face: the number of cubes along the horizontal (width) is 3, along the vertical (height) is 4. And along the depth (length) is 4? Wait, no, looking at the top layer: the number of cubes in the top layer: let's see, the top layer has 3 columns (width) and 4 rows (length)? Wait, no, the figure shows a rectangular prism where the number of cubes along length (l) = 4, width (w) = 3, height (h) = 4? Wait, no, maybe length is 4, width is 3, height is 4? Wait, no, let's check again.

Wait, the edge length of each small cube is 1 inch. So to find the volume of the rectangular prism, we use the formula \( V = l \times w \times h \), where \( l \), \( w \), \( h \) are the number of cubes along each dimension.

Looking at the figure:

- Along the length (depth): 4 cubes (since when we look at the depth, we can see 4 layers? Wait, no, the front face: the number of cubes along the height is 4 (4 rows vertically), along the width is 3 (3 columns horizontally), and along the length (depth) is 4 (4 columns going back). Wait, no, maybe length is 4, width is 3, height is 4? Wait, no, let's count the number of cubes in one layer (say the bottom layer). The bottom layer: how many cubes? If width is 3 and length is 4, then 3*4 = 12 cubes per layer. And there are 4 layers (height), so total cubes: 12*4 = 48? Wait, no, maybe I made a mistake. Wait, the figure: let's see the front face has 4 rows (height) and 3 columns (width). And the depth (length) has 4 columns. So length (l) = 4, width (w) = 3, height (h) = 4. Then volume \( V = 4 \times 3 \times 4 \)? Wait, no, that would be 48. Wait, but maybe I miscounted the dimensions.

Wait, another way: count the number of small cubes. Each small cube has a volume of \( 1 \times 1 \times 1 = 1 \) cubic inch. So if we can find the number of small cubes, that's the volume.

Looking at the figure:

- Number of cubes along length: 4
- Number of cubes along width: 3
- Number of cubes along height: 4

So total number of cubes = 4 (length) * 3 (width) * 4 (height) = 48. Wait, but let's check again. Wait, the front face: the number of cubes along the height is 4, along the width is 3. And along the depth (length) is 4. So 4*3*4 = 48. So the volume is 48 cubic inches.

Wait, maybe I made a mistake in the dimensions. Let's re - examine the figure. The figure shows a rectangular prism where:

- The number of cubes along the length (horizontal, depth) is 4.
- The number of cubes along the width (horizontal, front - back? No, width is left - right) is 3.
- The number of cubes along the height (vertical) is 4.

So using the formula for the volume of a rectangular prism \( V=l\times w\times h \), where \( l = 4 \) inches, \( w = 3 \) inches, \( h = 4 \) inches.

Step2: Calculate the volume

Substitute the values into the formula: \( V=4\times3\times4 \)

First, calculate \( 4\times3 = 12 \)

Then, \( 12\times4=48 \)

Answer:

\( 48 \) cubic inches (or \( 48\ \text{in}^3 \))