Question

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:

Step1: Count cubes in Box D

Count the number of unit - cubes in Box D. Assume the length \(l_D\), width \(w_D\), and height \(h_D\) of Box D. If we count, we find \(l_D = 5\), \(w_D=3\), \(h_D = 2\). The volume \(V_D=l_D\times w_D\times h_D=5\times3\times2 = 30\) cubic units.

Step2: Count cubes in Box E

Count the number of unit - cubes in Box E. Assume the length \(l_E\), width \(w_E\), and height \(h_E\) of Box E. If we count, we find \(l_E = 4\), \(w_E = 3\), \(h_E=3\). The volume \(V_E=l_E\times w_E\times h_E=4\times3\times3=36\) cubic units.

Step3: Compare volumes

Since \(V_D = 30\) and \(V_E = 36\), \(V_E>V_D\).

Step4: Add layer to Box D

Let's assume we add a layer of unit - cubes to Box D. If we add a layer on the top (keeping the length and width the same), the new height \(h_{D_{new}}=3\). The new volume \(V_{D_{new}}=l_D\times w_D\times h_{D_{new}}=5\times3\times3 = 45\) cubic units. Now, comparing the new volume of Box D (\(V_{D_{new}} = 45\)) with the volume of Box E (\(V_E = 36\)), we have \(V_{D_{new}}>V_E\).

Answer:

Initially, Box E has a greater volume. After adding a layer to Box D, the new Box D has a greater volume.