The total surface area of a right triangular prism of height 4 cm is 7

1.	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» (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.

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»

(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.