Question

a company makes japanese-style lunch boxes called bento boxes. each bento box is a cube with \\(\frac{1}{2}\\)-foot-long edges. the company ships the bento boxes in a container in the shape of a rectangular prism that measures \\(\frac{7}{2}\\) feet by \\(\frac{3}{2}\\) feet by \\(\frac{3}{2}\\) feet. a manager correctly finds the volume of the container using a formula. an employee double-checks the calculation by packing the container full of bento boxes. use the drop-down menus to explain how both volume calculations compare. click the arrows to choose an answer from each menu. the manager can find the volume of the container using the formula choose... . the employee correctly determines that the container can be filled with choose... bento boxes. he can find the volume of the container by multiplying the number of bento boxes that fill the container by the choose... of one bento box. the volume found by using the formula is choose... the volume of the total number of bento boxes in the container.

Explanation:

Step1: Volume formula for rectangular prism

The volume \( V \) of a rectangular prism is given by the formula \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height. For the container, \( l=\frac{7}{2} \), \( w = \frac{3}{2} \), \( h=\frac{3}{2} \). So the manager uses \( V=lwh \).

Step2: Number of bento boxes along each dimension

• Along the length: \( \frac{7}{2}\div\frac{1}{2}=7 \) (since each bento box has edge length \( \frac{1}{2} \) foot).
• Along the width: \( \frac{3}{2}\div\frac{1}{2}=3 \).
• Along the height: \( \frac{3}{2}\div\frac{1}{2}=3 \).
• Total number of bento boxes: \( 7\times3\times3 = 63 \).

Step3: Volume of one bento box

A bento box is a cube with edge length \( s=\frac{1}{2} \) foot. The volume of a cube is \( V_{cube}=s^3 \), so volume of one bento box is \( (\frac{1}{2})^3=\frac{1}{8} \) cubic feet. The employee multiplies the number of bento boxes (63) by the volume of one bento box (\( \frac{1}{8} \)) to get the container's volume.

Step4: Compare the two volumes

The volume from the formula: \( V=\frac{7}{2}\times\frac{3}{2}\times\frac{3}{2}=\frac{63}{8} \) cubic feet. The volume from bento boxes: \( 63\times\frac{1}{8}=\frac{63}{8} \) cubic feet. So they are equal.

Answer:

1. The manager can find the volume of the container using the formula \( \boldsymbol{V = l \times w \times h} \) (where \( l \), \( w \), \( h \) are length, width, height of the rectangular prism).
2. The employee correctly determines that the container can be filled with \( \boldsymbol{63} \) bento boxes.
3. He can find the volume of the container by multiplying the number of bento boxes that fill the container by the \( \boldsymbol{volume} \) of one bento box.
4. The volume found by using the formula is \( \boldsymbol{equal\ to} \) the volume of the total number of bento boxes in the container.