Question

a right rectangular prism with square bases has a height of 20 centimeters and a volume of 800 cubic centimeters. which statements describe the prism? select three options. the prism is a cube. the diagonal of the base is 4√5 centimeters. the length of a side of the base is 20 centimeters. the area of a base is 40 square centimeters. the area of a lateral side between the bases is about 126.5 square centimeters.

Explanation:

Step1: Find the area of the base

The volume formula of a rectangular - prism is $V = Bh$, where $B$ is the area of the base and $h$ is the height. Given $V = 800$ cm³ and $h = 20$ cm. We can find $B$ by $B=\frac{V}{h}$. So, $B=\frac{800}{20}=40$ cm².

Step2: Find the side - length of the square base

Since the base is a square and the area of a square is $B = s^{2}$ (where $s$ is the side - length of the square), and $B = 40$ cm², then $s=\sqrt{40}=2\sqrt{10}$ cm.

Step3: Check each option

• Option 1: A cube has all sides equal. Here, the height is 20 cm and the side - length of the base is $2\sqrt{10}$ cm, so it is not a cube.
• Option 2: The diagonal of a square with side - length $s$ is $d = s\sqrt{2}$. Since $s = 2\sqrt{10}$ cm, then $d=2\sqrt{10}\times\sqrt{2}=4\sqrt{5}$ cm.
• Option 3: We found that $s = 2\sqrt{10}

eq20$ cm.

• Option 4: We already found that $B = 40$ cm².
• Option 5: The area of a lateral side between the bases is $A = sh$, where $s = 2\sqrt{10}$ cm and $h = 20$ cm. So $A=2\sqrt{10}\times20 = 40\sqrt{10}\approx40\times3.162 = 126.48\approx126.5$ cm².

Answer:

B. The diagonal of the base is $4\sqrt{5}$ centimeters.
D. The area of a base is 40 square centimeters.
E. The area of a lateral side between the bases is about 126.5 square centimeters.