Mensuration
- If the areas of three adjacent faces of a rectangular box which meet in a corner are 12 cm2, 15 cm2 and 20 cm2 respectively, then the volume of the box is
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Let the length of rectangular box be l cm.
Width = b cm.
Height = h cm.
According to the question,
lb = 12 sq. cm. bh = 15 sq. cm.
hl = 20 sq. cm.
On multiplying,
l² × b² × h² = 12 × 15 × 20
∴ Volume of box = √12 × 15 × 20
= √3600 = 60 cu. cm.Correct Option: C
Let the length of rectangular box be l cm.
Width = b cm.
Height = h cm.
According to the question,
lb = 12 sq. cm. bh = 15 sq. cm.
hl = 20 sq. cm.
On multiplying,
l² × b² × h² = 12 × 15 × 20
∴ Volume of box = √12 × 15 × 20
= √3600 = 60 cu. cm.
- A cone, a cylinder and a hemisphere stand on equal bases and have equal heights. The ratio of their volumes is
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Ratio of volumes = cone : cylinder : hemi-sphere
= 1 πr²h : πr²h : 2 πr³ 3 3 = 1 πr³ : πr³ : 2 πr³ 3 3
[∵ r = h]= 1 : 1 : 2 = 1 : 3 : 2 3 3 Correct Option: C
Ratio of volumes = cone : cylinder : hemi-sphere
= 1 πr²h : πr²h : 2 πr³ 3 3 = 1 πr³ : πr³ : 2 πr³ 3 3
[∵ r = h]= 1 : 1 : 2 = 1 : 3 : 2 3 3
- The ratio of volumes of two cubes is 8 : 125. The ratio of their surface areas is
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Ratio of the volumes of cubes = 8 125 ⇒ l1³ = 8 ⇒ l1 = 2 l2³ 125 l2 25 ∴ Ratio of their total surface areas = 6l1² = l1² = 4 6l2² l2² 25 Correct Option: A
Ratio of the volumes of cubes = 8 125 ⇒ l1³ = 8 ⇒ l1 = 2 l2³ 125 l2 25 ∴ Ratio of their total surface areas = 6l1² = l1² = 4 6l2² l2² 25
- A spherical ball of radius 1 cm is dropped into a conical vessel of radius 3 cm and slant height 6 cm. The volume of water (in cm³), that can just immerse the ball, is
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Volume of water = 4 πr³ + 1 × 4 πr³ = 5 πr³ = 5 π cube cm. 3 4 3 3 3 Correct Option: A
Volume of water = 4 πr³ + 1 × 4 πr³ = 5 πr³ = 5 π cube cm. 3 4 3 3 3
- Assume that a drop of water is spherical and its diameter is onetenth of a cm. A conical glass has a height equal to the diameter of its rim. If 32000 drops of water fill the glass completely, then the height of the glass (in cm.) is
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Volume of one drop of water = 4 πr³ 3 = 4 × π × 
1 
³ cube cm. 3 20
&there; Volume of 32000 drops of water= 4π × 32000 cubic cm. = 16π cu.cm. 3 20 × 20 × 20 3 ∴ Volume of glass = 1 πR²H; Here, R = H 3 2 ∴ 1 π H ² H = 16π = 16 3 2 3 ⇒ H³ = 16 4
⇒ H³ = 64
∴ H = 64/3 = 4 cm.Correct Option: C
Volume of one drop of water = 4 πr³ 3 = 4 × π × 1 ³ cube cm. 3 20
&there; Volume of 32000 drops of water= 4π × 32000 cubic cm. = 16π cu.cm. 3 20 × 20 × 20 3 ∴ Volume of glass = 1 πR²H; Here, R = H 3 2 ∴ 1 π H ² H = 16π = 16 3 2 3 ⇒ H³ = 16 4
⇒ H³ = 64
∴ H = 64/3 = 4 cm.
