Mensuration
- The length, breadth and height of a cuboid are in the ratio 3 : 4 : 6 and its volume is 576 cm³. The whole surface of the cuboid is
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Length of parallelopiped = 3x cm
breadth = 4x cm
and height = 6x cm.
∴ Its volume = 576 cu.cm.
⇒ 3x × 4x × 6x = 576
⇒ 72x³ = 576⇒ x³ = √576 = 8 72
⇒ x = ³√8 = 2
∴ Total surface area = 2(l × b + b × h + h × l )
= 2(3x × 4x + 4x × 6x + 6x × 3x)
= 2(12x² + 24x2 + 18x2)
= 108 x²
= 108 × 2² = 108 × 4
= 432 sq. cm.Correct Option: C
Length of parallelopiped = 3x cm
breadth = 4x cm
and height = 6x cm.
∴ Its volume = 576 cu.cm.
⇒ 3x × 4x × 6x = 576
⇒ 72x³ = 576⇒ x³ = √576 = 8 72
⇒ x = ³√8 = 2
∴ Total surface area = 2(l × b + b × h + h × l )
= 2(3x × 4x + 4x × 6x + 6x × 3x)
= 2(12x² + 24x2 + 18x2)
= 108 x²
= 108 × 2² = 108 × 4
= 432 sq. cm.
- If the sum of the dimensions of a rectangular parallelopiped is 24 cm and the length of the diagonal is 15 cm, then the total surface area of it is
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Let, length = a cm.
breadth = b cm.
height = c cm.
∴ a + b + c = 24 --- (i)
and √a² + b² + c² = 15
⇒ a² + b² + c² = 15 × 15 = 225 - - - (ii)
∴ (a + b +c)² = a² + b² + c² + 2 (ab + bc + ca)
⇒ 24² = 225 + 2 (ab + bc + ca)
⇒ 576 = 225 + 2 (ab + bc + ca)
⇒ 2 (ab + bc + ca) = 576 – 225 = 351 sq.cm. = Total surface areaCorrect Option: C
Let, length = a cm.
breadth = b cm.
height = c cm.
∴ a + b + c = 24 --- (i)
and √a² + b² + c² = 15
⇒ a² + b² + c² = 15 × 15 = 225 - - - (ii)
∴ (a + b +c)² = a² + b² + c² + 2 (ab + bc + ca)
⇒ 24² = 225 + 2 (ab + bc + ca)
⇒ 576 = 225 + 2 (ab + bc + ca)
⇒ 2 (ab + bc + ca) = 576 – 225 = 351 sq.cm. = Total surface area
- Deepali makes a model of a cylindrical kaleidoscope for her science project. She uses a chart paper to make it. If the length of the kaleidoscope is 25 cm and radius 3-5 cm, the area of the paper she used, in square cm, is (taking π = 22/7)
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Required area = 2πrh
= 2 × 22 × 3.5 × 25 = 550 sq.cm. 7 Correct Option: B
Required area = 2πrh
= 2 × 22 × 3.5 × 25 = 550 sq.cm. 7
- The ratio of the length and breadth of a rectangular parallelopiped is 5 : 3 and its height is 6 cm. If the total surface area of the parallelopiped be 558 sq. cm, then its length in dm is
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Length = 5x cm Breadth = 3x cm
Total surface area of parallelopiped = 2(l × b + b × h + h × l)
= 2(5x × 3x + 3x × 6 + 6 × 5x)
= 2(15x² + 18x + 30x)
= 2 (15x² + 48x)
∴ 2(15x² + 48x) = 558⇒ 15x² + 48x = 558 = 279 2
⇒ 5x² + 16x = 93
⇒ 5x² + 16x – 93 = 0
⇒ 5x² + 31x – 15x – 93 = 0
⇒ x (5x + 31) – 3 (5x + 31) = 0
⇒ (x – 3) (5x + 31) = 0
⇒ x = 3
∴ Length = 5x = 5 × 3 = 15 cm
= 1.5 dmCorrect Option: B
Length = 5x cm Breadth = 3x cm
Total surface area of parallelopiped = 2(l × b + b × h + h × l)
= 2(5x × 3x + 3x × 6 + 6 × 5x)
= 2(15x² + 18x + 30x)
= 2 (15x² + 48x)
∴ 2(15x² + 48x) = 558⇒ 15x² + 48x = 558 = 279 2
⇒ 5x² + 16x = 93
⇒ 5x² + 16x – 93 = 0
⇒ 5x² + 31x – 15x – 93 = 0
⇒ x (5x + 31) – 3 (5x + 31) = 0
⇒ (x – 3) (5x + 31) = 0
⇒ x = 3
∴ Length = 5x = 5 × 3 = 15 cm
= 1.5 dm
- The base of a prism is a right angled triangle with two sides 5 cm and 12 cm. The height of the prism is 10 cm. The total surface area of the prism is
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Hypotenuse of base = √5² + 12²
= √25 + 144 = √169 = 13 cm
∴ Surface area = h (a+ b + c)
= 10 (5 + 12 + 13) = 300 sq.cm.Area of base = 1 × 5 × 12 2
= 30 sq.cm.
∴ Total surface area of lateral surfaces = 300 + 30 = 330 sq.cm.Correct Option: C
Hypotenuse of base = √5² + 12²
= √25 + 144 = √169 = 13 cm
∴ Surface area = h (a+ b + c)
= 10 (5 + 12 + 13) = 300 sq.cm.Area of base = 1 × 5 × 12 2
= 30 sq.cm.
∴ Total surface area of lateral surfaces = 300 + 30 = 330 sq.cm.
