Question

If the two prisms have equal heights and volumes, what is the length of the x in Prism II?

Answer 1

The volume of a prism is given by

$V = A_b\cdot h$

Where $A_b$ is the base are and $h$ is the height.

So, if two prisms have the same height and the same volume, they must have the same base area.

The first base is an equilateral triangle, which means that the height is given by

$h = \dfrac{l\sqrt{3}}{2}$

So, the area is

$A = 8\cdot\dfrac{8\sqrt{3}}{2} = 32\sqrt{3}$

The area of the second base is simply the product of the dimensions, so it is $6.9x$

Since the two areas must be the same, we have

$32\sqrt{3}=6.9x \iff x = \dfrac{32\sqrt{3}}{6.9} \approx 8.03270$