Question

1. exploration

a. han says that the volume of this rectangular prism is 50 times as great as a 2 - inch cube. do you agree with han? explain or show your reasoning
b. han says that he can fit 50 2 - inch cubes in this rectangular prism. do you agree with han? explain or show your reasoning.

1. exploration

two common sizes of shipping boxes are 5 inches by 6 inches by 15 inches and 11 inches by 6 inches by 15 inches. which size box would you choose to ship the student workbooks for your math class? explain or show your reasoning.

Explanation:

Step1: Calculate volume of rectangular prism

The volume formula for a rectangular prism is $V = l\times w\times h$. Given $l = 10$ in, $w = 8$ in and $h = 5$ in, so $V_{prism}=10\times8\times5=400$ in³.

Step2: Calculate volume of 2 - inch cube

The volume formula for a cube is $V = s^3$. Given $s = 2$ in, so $V_{cube}=2^3 = 8$ in³.

Step3: Compare volumes for part a

Find the ratio of the volume of the prism to the volume of the cube. $\frac{V_{prism}}{V_{cube}}=\frac{400}{8}=50$. So the volume of the rectangular prism is 50 times as great as the 2 - inch cube.

Step4: Calculate number of 2 - inch cubes that fit for part b

The length of the prism $l = 10$ in can fit $\frac{10}{2}=5$ 2 - inch cubes along the length. The width $w = 8$ in can fit $\frac{8}{2}=4$ 2 - inch cubes along the width. The height $h = 5$ in can fit $\frac{5}{2}=2.5$ (but we take the whole - number part, so 2) 2 - inch cubes along the height. The total number of 2 - inch cubes that can fit is $5\times4\times2=40$. So 50 2 - inch cubes cannot fit.

Answer:

a. Yes. The volume of the rectangular prism is 400 in³ and the volume of the 2 - inch cube is 8 in³, and $\frac{400}{8}=50$.
b. No. The maximum number of 2 - inch cubes that can fit is 40.