In the diagram, ABCDEF is a triangular prism. < ABC = < DEF = 90°, /AB/ = 24 cm, /BC/ = 7 cm and /CD/ = 40 cm. Calculate : (a) /AC/ ; (b) the total surface ...

Answer Details

The solid is a triangular prism with congruent right-angled triangular ends \(ABC\) and \(DEF\) (\(\angle ABC = \angle DEF = 90^\circ\)). The given lengths are \(|AB| = 24\) cm, \(|BC| = 7\) cm, and the prism length \(|CD| = 40\) cm.

(a) Length AC. Triangle \(ABC\) is right-angled at \(B\), so by Pythagoras:

\[ |AC| = \sqrt{|AB|^2 + |BC|^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25\ \text{cm} \]

(b) Total surface area. The prism has two triangular ends and three rectangular faces. The prism length is \(40\) cm.

The two triangular ends:

\[ 2 \times \left(\tfrac{1}{2}\times 24 \times 7\right) = 2 \times 84 = 168\ \text{cm}^2 \]

The three rectangles have widths equal to the sides of the triangle \((24, 7, 25)\) and common length \(40\):

\[ (24 + 7 + 25)\times 40 = 56 \times 40 = 2240\ \text{cm}^2 \]

Total surface area:

\[ 168 + 2240 = 2408\ \text{cm}^2 \]

Answers: (a) \(|AC| = 25\) cm; (b) total surface area \(= 2408\ \text{cm}^2\).