Question
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 :
Answer: Option B
Let the length of base be 3a cm and breadth be 2a cm
Total surface area of prism :
= [Perimeter of base × height] + [2 × Area of base]
= [2 (3a + 2a) × 12 + 2 × 3a × 2a] sq.cm
= (120a + 12a2) sq.cm
According to the question,
120a + 12a2 = 288
⇒ a2 + 10a = 24
⇒ a2 + 10a - 24 = 0
⇒ a2 + 12a - 2a - 24 = 0
⇒ a (a + 12) - 2 (a + 12) = 0
⇒ (a - 2)(a + 12) = 0
⇒ a = 2 because a $$ \ne $$ -12
∴ Volume of prism :
= Area of base × Height
= (3a × 2a × 12)cu.cm
= 72a2 cu.cm
= (72 × 2 × 2)cu.cm
= 288 cu.cm
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Let the length of base be 3a cm and breadth be 2a cm
Total surface area of prism :
= [Perimeter of base × height] + [2 × Area of base]
= [2 (3a + 2a) × 12 + 2 × 3a × 2a] sq.cm
= (120a + 12a2) sq.cm
According to the question,
120a + 12a2 = 288
⇒ a2 + 10a = 24
⇒ a2 + 10a - 24 = 0
⇒ a2 + 12a - 2a - 24 = 0
⇒ a (a + 12) - 2 (a + 12) = 0
⇒ (a - 2)(a + 12) = 0
⇒ a = 2 because a $$ \ne $$ -12
∴ Volume of prism :
= Area of base × Height
= (3a × 2a × 12)cu.cm
= 72a2 cu.cm
= (72 × 2 × 2)cu.cm
= 288 cu.cm
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