Question

the solid below is made from cubes. find its volume.

Explanation:

Step1: Determine the number of cubes along each dimension

Looking at the figure, along the length: there are 5 cubes. Along the width: there are 2 cubes. Along the height: there are 3 cubes.

Step2: Calculate the volume of one small cube

Since the side length of the small cube corresponding to the "1 m" mark (assuming the base unit for each cube's edge is such that the total length for 2 cubes is 1 m? Wait, no, maybe the side length of each small cube is \( \frac{1}{2} \) m? Wait, no, let's re - examine. Wait, the figure shows that the length of 2 cubes is 1 m? Wait, no, the label "1 m" is under the base, maybe the length of the base (along the width) is 1 m with 2 cubes, so each cube has a side length of \( \frac{1}{2} \) m? Wait, no, maybe I misread. Wait, actually, let's count the number of cubes: length = 5, width = 2, height = 3. So total number of cubes \( n=5\times2\times3 = 30 \). Now, if we assume that the side length of each small cube is \( s \), and from the diagram, the width (with 2 cubes) is 1 m, so \( 2s=1\), so \( s = \frac{1}{2}\) m. Then the volume of one small cube is \( V_{cube}=s^{3}=(\frac{1}{2})^{3}=\frac{1}{8}\text{ m}^3 \). But wait, that seems complicated. Wait, maybe the "1 m" is the length of 5 cubes? No, the diagram shows that the width (the side with 2 cubes) is labeled 1 m. So width \( w = 1\) m, number of cubes along width \( n_w=2 \), so side length of each cube \( s=\frac{1}{2}\) m. Length \( l \): number of cubes along length \( n_l = 5\), so \( l=5\times\frac{1}{2}=\frac{5}{2}\) m. Height \( h \): number of cubes along height \( n_h=3 \), so \( h = 3\times\frac{1}{2}=\frac{3}{2}\) m. Then volume \( V=l\times w\times h=\frac{5}{2}\times1\times\frac{3}{2}=\frac{15}{2}=7.5\)? No, that can't be. Wait, maybe the side length of each small cube is 1 unit, and the "1 m" is the length of 2 cubes, so each cube has a side length of \( \frac{1}{2}\) m. But the total number of cubes is \( 5\times2\times3 = 30 \). The volume of each small cube is \( (\frac{1}{2})^3=\frac{1}{8}\text{ m}^3 \). Then total volume \( V = 30\times\frac{1}{8}=\frac{15}{4}=3.75 \)? No, this is confusing. Wait, maybe I made a mistake. Wait, the problem says "the solid below is made from cubes" and we need to find its volume. Let's count the number of cubes: length = 5, width = 2, height = 3. So total number of cubes \( N=5\times2\times3 = 30 \). Now, if we assume that the side length of each small cube is \( \frac{1}{2}\) m (because the width with 2 cubes is 1 m), then the volume of one cube is \( (\frac{1}{2})^3=\frac{1}{8}\text{ m}^3 \). Then total volume \( V = 30\times\frac{1}{8}=\frac{15}{4}=3.75 \)? No, that's not matching. Wait, maybe the "1 m" is the side length of each small cube? But the width has 2 cubes, so that would make the width 2 m, but the label is 1 m. Wait, maybe the diagram is such that the length of 5 cubes is, say, \( L \), width of 2 cubes is 1 m, and height of 3 cubes is \( H \). So side length of each cube \( s=\frac{1}{2}\) m. Then volume of the solid is \( V=(5\times\frac{1}{2})\times1\times(3\times\frac{1}{2})=\frac{5}{2}\times1\times\frac{3}{2}=\frac{15}{4} = 3.75 \)? But the answer box has \( \square^3\text{ m} \), maybe the side length of each cube is \( \frac{1}{2}\) m, and we calculate the volume as the number of cubes times the volume of each cube. Wait, no, let's try another approach. Let's assume that the "1 m" is the length of the base (width) which has 2 cubes, so each cube has a side length of \( \frac{1}{2}\) m. Then the volume of one cube is \( (\frac{1}{2})^3=\frac{1}{8}\text{ m}^3 \). The number of cubes is \( 5\times2\times3=30 \). So total volume \( V = 30\times\frac{1}{8}=\frac{15}{4}=3.75 \text{ m}^3 \). Wait, but maybe I misinterpret the diagram. Alternatively, maybe the side length of each small cube is 1 m, but the width (with 2 cubes) is 1 m, so that's a contradiction. Wait, maybe the diagram is drawn such that the length is 5 units, width is 2 units, height is 3 units, and each unit cube has a volume of \( (\frac{1}{2})^3 \) because the width of 2 units is 1 m. So volume \( V=5\times2\times3\times(\frac{1}{2})^3=30\times\frac{1}{8}=\frac{15}{4} = 3.75 \text{ m}^3 \). But let's check again. Wait, maybe the "1 m" is the side length of each small cube, and the width has 2 small cubes, so the width is 2 m, but the label says 1 m. So there must be a misinterpretation. Wait, another way: the total volume of a rectangular prism is \( V = l\times w\times h \). From the figure, if we consider that the length (number of cubes along length) is 5, width (number of cubes along width) is 2, height (number of cubes along height) is 3. And if the width of 2 cubes is 1 m, then each cube has a side length of \( \frac{1}{2}\) m. So \( l = 5\times\frac{1}{2}=\frac{5}{2}\) m, \( w = 1\) m, \( h=3\times\frac{1}{2}=\frac{3}{2}\) m. Then \( V=\frac{5}{2}\times1\times\frac{3}{2}=\frac{15}{4}=3.75\text{ m}^3 \). But the answer box is expecting a number, maybe I made a mistake in the number of cubes. Wait, let's count the cubes: in the front face, there are 5 columns and 3 rows, so \( 5\times3 = 15 \) cubes, and there are 2 layers (width), so total cubes \( 15\times2=30 \). If each cube has a volume of \( (\frac{1}{2})^3=\frac{1}{8}\text{ m}^3 \), then total volume \( 30\times\frac{1}{8}=\frac{15}{4}=3.75 \text{ m}^3 \). But maybe the problem is that each small cube has a side length of \( \frac{1}{2}\) m, so volume of each cube is \( (\frac{1}{2})^3=\frac{1}{8}\text{ m}^3 \), and total number of cubes is \( 5\times2\times3 = 30 \), so total volume \( V = 30\times\frac{1}{8}=\frac{15}{4}=3.75 \text{ m}^3 \). But let's check the answer format. The answer box has \( \square^3\text{ m} \), so we need to put the number. Wait, maybe I misread the number of cubes. Wait, looking at the figure again: the length (horizontal) has 5 cubes, the width (depth) has 2 cubes, the height (vertical) has 3 cubes. So total number of cubes \( n = 5\times2\times3=30 \). Now, if the side length of each cube is \( s \), and from the label "1 m" under the base (width), which has 2 cubes, so \( 2s = 1\), so \( s=\frac{1}{2}\) m. Then volume of one cube \( v=s^3=(\frac{1}{2})^3=\frac{1}{8}\text{ m}^3 \). Then total volume \( V=n\times v = 30\times\frac{1}{8}=\frac{15}{4}=3.75 \text{ m}^3 \). But maybe the problem is simpler: maybe each small cube has a volume of \( (\frac{1}{2})^3 \), but the number of cubes is \( 5\times2\times3 = 30 \), so \( 30\times(\frac{1}{2})^3=30\times\frac{1}{8}=3.75 \). Alternatively, maybe the "1 m" is the length of 5 cubes, so \( s=\frac{1}{5}\) m, but then width with 2 cubes would be \( \frac{2}{5}\) m, which doesn't match the label. So the most reasonable is that the width of 2 cubes is 1 m, so \( s = \frac{1}{2}\) m, and total volume is \( 5\times2\times3\times(\frac{1}{2})^3=\frac{15}{4}=3.75 \text{ m}^3 \).

Answer:

\( 3.75 \)