Answer:
The formula for the volume of a triangular prism is given by:
V = (1/2)bhL
where b is the base length of the triangle, h is the height of the triangle, and L is the length of the prism (i.e., the distance between the two triangular faces).
We are given the area of the rectangular surface of the prism, which is the product of the lengths of two adjacent sides of the rectangle, and we know that this area is equal to 864 cm^2. Therefore, we can use the Pythagorean theorem to find the length of the third side of the rectangle, which is also the base length of the triangular face:
17^2 = 9^2 + 10^2
289 = 81 + 100
289 = 181
So, the base length of the triangular face is 8 cm. Now we can plug in the known values into the formula for the volume of the prism:
864 = (1/2)(8 cm)(h)(L)
We are not given the length of the prism, L, but we know that it is equal to the base length of the triangle, which is 8 cm. Therefore, we can simplify the equation to:
864 = 32h
Solving for h, we get:
h = 864/32
h = 27
So the height of the prism is 27 cm.