Use the image to answer the question.
An illustration shows a two-dimensional net of a triangular prism with all of its sides open and visible. The net appears as three horizontal rectangles joined one on top of another. All 4 sides of the middle rectangle are drawn with dashed lines. The length of the rectangles is marked with 3 question marks. The width of the middle rectangle is 24 units. The middle rectangle has a triangle adjoining its left side. The other two sides of the adjoining triangle measure 25 units for the hypotenuse and 7 units for the base. An unmarked triangle adjoins the right side of the bottom rectangle.
The surface area of the triangular prism shown is 5,768 square units. Other dimensions are also shown on the net. What is the height of the prism?
A:34.3 units
B:100 units
C:2,400 units
D:84 units
First, we need to find the area of the two triangles on the net. The area of the triangle adjoining the left side of the middle rectangle is (1/2)(7)(24) = 84 square units. The area of the triangle adjoining the right side of the bottom rectangle can be found by using the Pythagorean theorem: 7² + h² = 25², where h is the height of the triangle (and also the height of the prism). Solving for h, we get h = 24 units. The area of this triangle is (1/2)(7)(24) = 84 square units.
Next, we can use the surface area formula for a triangular prism: SA = 2B + Ph, where B is the area of the triangular base, P is the perimeter of the base, and h is the height of the prism. We already found B (the area of each triangular base) to be 84 square units. The perimeter of the rectangular base is 2(???) + 2(24) = 2(???) + 48, but we don't know the length of the rectangles yet. However, we do know that the surface area is 5,768 square units. Substituting the known values into the surface area formula, we get:
5768 = 2(84) + (2??? + 48)h
5768 = 168 + (2??? + 48)h
5596 = (2??? + 48)h
116.583... = 2??? + 48
68.583... = 2???
34.291... = ???
Therefore, the height of the prism (and the unmarked triangle on the net) is approximately 34.3 units, which is answer choice A.