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Base of a right prism is a rectangle, the ratio of whose length and breadth is 3 : 2. If the height of the prism is 12 cm and total surface area is 288 sq. cm., the volume of the prism is :
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- 291 cm³
- 288 cm³
- 290 cm³
- 286 cm³
- 291 cm³
Correct Option: B
Length of base = 3x cm and breadth = 2x cm (let)
Total surface area of prism = perimeter of base × height + 2 × area of base
= [2(3x + 2x) × 12 + 2 × 3x × 2x] sq. cm.
= (120x + 12x2) sq. cm.
According to the question,
120x + 12x² = 288
⇒ x² + 10x= 24
⇒ x² + 10x – 24 = 0
⇒ x² + 12x – 2x – 24 = 0
⇒ x (x + 12) – 2(x + 12) = 0
⇒ (x – 2) (x +12) = 0
⇒ x = 2 because x ≠ –12
∴ Volume of prism = Area of base × height = (3x × 2x × 12) cu. cm.
= 72x² = (72 × 2 × 2) cu. cm.
= 288 cu. cm.
