Question

The total surface area of a right triangular prism of height 4 cm is 72√3 cm². If the base of the prism is an equilateral triangle, find its volume. 

(a) 36√3 3 cm³

(b) 42√3 cm³

(c) 48√3 cm³

(d) 54√3 cm³

Answer 1

(c) \(48\sqrt3\) cm².

Let each side of the base of the prism be a cm. 

Total surface area = \(72\sqrt3\) cm³ 

⇒ (Perimeter of the base × Height) + 2 (Area of base) = \(72\sqrt3\)

⇒ 3a x 4 + 2 \(\big(\frac{\sqrt3}{4}a^2\big)\) = \(72\sqrt3\)

⇒ \(\sqrt3a^2+24a-144\sqrt3=0\) ⇒ \(a^2+8\sqrt3a-144=0\)

⇒ \((a+12\sqrt3)(a-4\sqrt3)=0\)

⇒ \(a=-12\sqrt3\) or \(4\sqrt3\)  ⇒ a = \(4\sqrt3\)  as a > 0

∴ Volume of the prism = Area of the base × Height 

= \(\frac{\sqrt3}{4}\times(4\sqrt3)^2\times\,4cm^2=48\sqrt3\) cm².