Question

a right prism has a rhombus as a base. the height of the prism is 6 inches and the volume is 144 cubic inches. which could be the lengths of the diagonals of the rhombus? 2 in by 12 in. 4 in by 11 in. 6 in by 8 in. 8 in by 9 in.

Explanation:

Step1: Recall volume formula for prism

The volume formula for a prism is $V = Bh$, where $B$ is the base - area and $h$ is the height. Given $V = 144$ cubic inches and $h=6$ inches.

Step2: Calculate the base - area

We can find the base - area $B$ by rearranging the formula: $B=\frac{V}{h}$. Substituting the given values, $B=\frac{144}{6}=24$ square inches.

Step3: Recall area formula for rhombus

The area formula for a rhombus is $A=\frac{1}{2}d_1d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.

Step4: Check each option

For option A: If $d_1 = 2$ in and $d_2 = 12$ in, then $A=\frac{1}{2}\times2\times12 = 12$ square inches.
For option B: If $d_1 = 4$ in and $d_2 = 11$ in, then $A=\frac{1}{2}\times4\times11 = 22$ square inches.
For option C: If $d_1 = 6$ in and $d_2 = 8$ in, then $A=\frac{1}{2}\times6\times8 = 24$ square inches.
For option D: If $d_1 = 8$ in and $d_2 = 9$ in, then $A=\frac{1}{2}\times8\times9 = 36$ square inches.

Answer:

C. 6 in by 8 in