Question

mei uses unit cubes to measure the volume of the box shown. she thinks the volume of the box is 8 cubic units. do you agree? explain.

Explanation:

Step1: Analyze the box's dimensions

From the diagram, we can see that the box (a rectangular prism) has a length, width, and height. Let's assume the number of unit cubes along each dimension. Looking at the visible unit cubes, we can infer the dimensions. Let's say the length has \( l \) unit cubes, width \( w \), and height \( h \). From the diagram, if we count the visible layers, let's assume length \( l = 4 \), width \( w = 2 \), height \( h = 1 \)? Wait, no, maybe better to count the number of unit cubes. Wait, actually, the volume of a rectangular prism is \( V = l \times w \times h \), where \( l \), \( w \), \( h \) are the number of unit cubes along each edge. Wait, looking at the diagram, maybe the box has dimensions: let's see, the front face has 4 unit cubes vertically? Wait, no, the diagram shows a box with some unit cubes inside. Wait, maybe the correct dimensions are length \( = 4 \), width \( = 2 \), height \( = 1 \)? No, wait, maybe I misread. Wait, actually, the volume of a rectangular prism is calculated by multiplying the number of unit cubes along each side. Let's look at the diagram again. The box appears to have a length of 4 unit cubes, a width of 2 unit cubes, and a height of 1 unit cube? No, that would be \( 4 \times 2 \times 1 = 8 \)? Wait, no, maybe the height is 1? Wait, no, maybe the dimensions are length \( = 4 \), width \( = 2 \), height \( = 1 \), so volume \( 4 \times 2 \times 1 = 8 \). Wait, but maybe I made a mistake. Wait, no, let's think again. The volume of a rectangular prism is \( V = l \times w \times h \), where \( l \), \( w \), \( h \) are the number of unit cubes along the length, width, and height. If the box can be filled with unit cubes, and from the diagram, if we see that along the length there are 4 unit cubes, along the width 2, and along the height 1, then \( V = 4 \times 2 \times 1 = 8 \). But wait, maybe the height is more? Wait, the diagram shows a box with a front face that has 4 unit cubes (vertically) and 2 unit cubes (horizontally)? No, maybe the correct dimensions are length \( = 4 \), width \( = 2 \), height \( = 1 \), so volume is \( 4 \times 2 \times 1 = 8 \). Wait, but maybe I'm wrong. Wait, no, let's check the problem again. Mei thinks the volume is 8 cubic units. Let's confirm the dimensions. If the box is a rectangular prism, and the number of unit cubes along length is \( l \), width \( w \), height \( h \), then volume \( V = l \times w \times h \). From the diagram, if we count the unit cubes: let's say along the length (the longer side) there are 4 unit cubes, along the width (the side perpendicular to length) there are 2 unit cubes, and along the height (the vertical side) there is 1 unit cube. Then \( V = 4 \times 2 \times 1 = 8 \). So yes, the volume would be 8 cubic units. So we agree with Mei.

Step2: Calculate the volume

The formula for the volume of a rectangular prism (box) using unit cubes is \( V = \text{number of unit cubes along length} \times \text{number of unit cubes along width} \times \text{number of unit cubes along height} \). From the diagram, let's assume the length has 4 unit cubes, width has 2 unit cubes, and height has 1 unit cube. Then:
\[
V = 4 \times 2 \times 1 = 8
\]
So the volume is 8 cubic units, which matches Mei's thought.

Answer:

Yes, I agree. The volume of a rectangular prism (box) is calculated by multiplying the number of unit cubes along its length, width, and height. If the box has dimensions (number of unit cubes) \( 4 \times 2 \times 1 \), then the volume is \( 4 \times 2 \times 1 = 8 \) cubic units, so Mei is correct.