Question

4 jenn noticed that she can fit two juice boxes side by side on the bottom of this box. she can make two more layers like the one shown to fill the box. using a juice box as a measure of volume, what is the volume of the larger box?

solution

5 a box is 2 inches long, 1 inch wide, and 6 inches tall. what is the relationship between the volume of this box and the one in problem 4? tell how you kn

6 box d and box e are made from unit cubes of the same size. which has a greater volume, box d or box e? explain.

add a layer to box d and compare the volume of the new box d to the volume of box e

Explanation:

Problem 4

Step1: Determine number of juice boxes per layer

From the problem, we know that on the bottom layer, we can fit 2 juice boxes side by side. Looking at the diagram (a cube - like box with layers), we can assume the base has 2 juice boxes in one dimension and 1 in the other (since it's side - by - side, maybe a 2×1 base for the layer).

Step2: Determine number of layers

She can make two more layers like the one shown. So the number of layers is the initial layer plus two more layers. So total layers \(= 1+2 = 3\) layers.

Step3: Calculate total number of juice boxes (volume)

The volume of the larger box, when measured by juice boxes, is the number of juice boxes per layer times the number of layers. The number of juice boxes per layer is \(2\times1=2\) (assuming the base is 2 juice boxes long and 1 juice box wide). Then total number of juice boxes \(=2\times3 = 6\)? Wait, maybe the base is 2 juice boxes in length and 1 in width, and the height (number of layers) is 3. Wait, maybe the initial layer is 2 juice boxes (side - by - side), and then we have 3 layers (1 original + 2 more). So total juice boxes \(=2\times3=6\)? Wait, maybe the diagram shows a 2 (length) ×1 (width) ×3 (height) arrangement. So volume (in juice boxes) is \(2\times1\times3 = 6\). Wait, but maybe the initial layer is 2 juice boxes (so length = 2, width = 1), and the number of layers is 3 (since 1 layer + 2 more layers). So total volume is \(2\times1\times3=6\) juice boxes.

Step1: Calculate volume of the given box

The formula for the volume of a rectangular box is \(V=l\times w\times h\), where \(l\) is length, \(w\) is width and \(h\) is height. Given \(l = 2\) inches, \(w = 1\) inch, \(h=6\) inches. So \(V = 2\times1\times6=12\) cubic inches.

Step2: Compare with problem 4's volume

From problem 4, we found the volume of the larger box (in juice - box units) was 6. Wait, maybe there was a miscalculation in problem 4. Wait, maybe the diagram for problem 4 is a box where the base has 2 juice boxes (length = 2), width = 1 juice box, and height = 3 layers (so 3 juice boxes tall). Then volume is \(2\times1\times3 = 6\) juice - box units. But the box in problem 5 has volume \(2\times1\times6 = 12\) cubic inches. So the volume of the box in problem 5 is twice the volume of the box in problem 4. Because \(12\div6 = 2\). We know this because we calculated the volume of each box using the formula for the volume of a rectangular prism (\(V = l\times w\times h\)) and then compared the two volumes.

Step1: Recall volume of a box made of unit cubes

The volume of a box made from unit cubes is equal to the number of unit cubes it contains. The number of unit cubes is calculated by multiplying the number of unit cubes along the length, width, and height (\(V=l\times w\times h\)).

Step2: Assume dimensions for Box D and Box E (from the hand - written work)

From the hand - written work, for Box D: maybe \(l = 3\), \(w = 1\), \(h = 5\) (but the hand - written calculation is \(3\times1\times5 = 15\))? Wait, the hand - written work has \(3\times1\times5 = 15\) and for Box E: \(4\times4\times1=16\)? Wait, maybe the correct dimensions: Let's assume that Box D has dimensions (number of unit cubes along each side) \(3\times1\times5\) (volume \(=3\times1\times5 = 15\)) and Box E has dimensions \(4\times4\times1\) (volume \(=4\times4\times1 = 16\))? Wait, no, maybe the hand - written work has some errors. Wait, the key idea is that if we calculate the number of unit cubes (volume) for each box:
Suppose Box D has length \(l_D\), width \(w_D\), height \(h_D\) and Box E has length \(l_E\), width \(w_E\), height \(h_E\). Then \(V_D=l_D\times w_D\times h_D\) and \(V_E=l_E\times w_E\times h_E\). From the hand - written work, if for Box D: \(3\times1\times5 = 15\) and for Box E: \(4\times4\times1=16\) (or maybe other dimensions). Wait, the hand - written answer has a box with 'D'? No, wait the hand - written work has some calculations: \(3\times1\times5 = 15\) and \(4\times4\times1 = 16\). So if Box D has volume 15 and Box E has volume 16, then Box E has a greater volume because \(16>15\). The volume of a box made of unit cubes is the product of the number of unit cubes along each edge, so we calculate the volume for each box and compare the two products.

Answer:

The volume of the larger box is 6 juice - box units.

Problem 5