5355507
A rectangular prism is \(6\,\text{cm}\) long. Each end is a \(3\,\text{cm} \times 4\,\text{cm}\) rectangle whose diagonal is \(5\,\text{cm}\). The prism is cut by a plane that passes through the diagonal of each end and is parallel to the prism's length. One resulting solid is shown.
a) What shape is the cross-section made by the cut?
b) What are the dimensions of the cross-section?
c) Find the area of the cross-section.
d) What exact type of solid is each resulting piece?
Hints
- Trace where the cutting plane meets each end of the prism.
- One cross-section dimension comes from the end-face diagonal, and the other comes from the prism's length.
- Identify the shapes of the two parallel bases of each resulting solid.
Solution
1. The cut follows a line segment across each end and extends straight along the length of the prism, so the cross-section is a rectangle.
2. One side of the cross-section is the \(5\,\text{cm}\) diagonal of an end. The other side is the \(6\,\text{cm}\) length of the prism, so its dimensions are \(5\,\text{cm} \times 6\,\text{cm}\).
3. Its area is \(5 \cdot 6=30\,\text{cm}^2\).
4. Each end of a resulting piece is a triangle, so each piece is a triangular prism.
Answer
a) A rectangle
b) \(5\,\text{cm} \times 6\,\text{cm}\)
c) \(30\,\text{cm}^2\)
d) A triangular prism
