A prism of length \(13\) cm has a volume of \(156\) cm\(^3\). The cross-section is the triangle shown in the diagram, with base \(8\) cm. Work out the perpe...
Question 1 Report
A prism of length \(13\) cm has a volume of \(156\) cm\(^3\).
The cross-section is the triangle shown in the diagram, with base \(8\) cm.
Work out the perpendicular height of this triangle.
Answer Details
This runs the prism volume formula backwards. Since volume equals cross-sectional area times length, dividing the volume by the length recovers the area of the cross-section, and the triangle area formula then gives the missing height.
- Area of the triangular cross-section: \[ \frac{156}{13} = 12 \text{ cm}^2 \] [M1]
- For the triangle, \(\frac{1}{2} \times 8 \times h = 12\), so \(4h = 12\) and the perpendicular height is \(h = 3\) cm [A1].
Working in two clear stages, volume to area and then area to height, keeps the units honest: \(156\) cm\(^3\) divided by \(13\) cm gives cm\(^2\), and dividing that by a length in cm gives cm. A frequent error is to divide \(156\) by \(8\) or by \(13 \times 8\) in one step without halving, forgetting the \(\frac{1}{2}\) in the triangle formula. Substituting back is a quick check: \(\frac{1}{2} \times 8 \times 3 \times 13 = 156\) cm\(^3\).
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