Mensuration
- A right circular cylinder and a cone have equal base radius and equal height. If their curved surfaces are in the ratio 8 : 5, then the radius of the base to the height are in the ratio :
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Let Radius of the base = r units and height = h units
⇒ Curvedsurfaceof cylinder Curvedsurfaceof cone = 2πrh πrl ⇒ 8 = 2h 5 l ⇒ 4 = h 5 √h² + r² ⇒ 16 = h² 25 h² + r² ⇒ h² + r² = 25 h² 16 ⇒ 1 + r² = 25 h² 16 ⇒ r² = 25 - 1 = 9 h² 16 16 ⇒ r = 3 h 4
or 3 : 4Correct Option: C
Let Radius of the base = r units and height = h units
⇒ Curvedsurfaceof cylinder Curvedsurfaceof cone = 2πrh πrl ⇒ 8 = 2h 5 l ⇒ 4 = h 5 √h² + r² ⇒ 16 = h² 25 h² + r² ⇒ h² + r² = 25 h² 16 ⇒ 1 + r² = 25 h² 16 ⇒ r² = 25 - 1 = 9 h² 16 16 ⇒ r = 3 h 4
or 3 : 4
- A circus tent is cylindrical up to a height of 3 m and conical above it. If its diameter is 105m and the slant height of the conical part is 63 m, then the total area of the canvas required to make the tent is take ( π = 22/7)
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Total area of the canvas = 2πrh + πrl = πr (2h + l)
= 22 × 105 (2 × 3 + 63) 7 3 = 22 × 105 × 69 7 3
= 11385 sq.metreCorrect Option: A
Total area of the canvas = 2πrh + πrl = πr (2h + l)
= 22 × 105 (2 × 3 + 63) 7 3 = 22 × 105 × 69 7 3
= 11385 sq.metre
- A sphere and a cylinder have equal volume and equal radius. The ratio of the curved surface area of the cylinder to that of the sphere is
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Volume of sphere Volume of cylinder = 4 πr³ = 1 ⇒ = 3 r 3 πr²h h 4 ∴ Curved surface area of cylinder Surface area of sphere = 2πrh = h = 1 × 4 = 2 4πr² 2r 2 3 3
or 2 : 3Correct Option: B
Volume of sphere Volume of cylinder = 4 πr³ = 1 ⇒ = 3 r 3 πr²h h 4 ∴ Curved surface area of cylinder Surface area of sphere = 2πrh = h = 1 × 4 = 2 4πr² 2r 2 3 3
or 2 : 3
- A right circular cylinder just encloses a sphere of radius r. The ratio of the surface area of the sphere and the curved surface area of the cylinder is
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Let Height of the cylinder = 2r
Curved surface area of the cylinder = 2πRH
∴ Required ratio = 4πr² : 2π × r × 2r = 1 : 1Correct Option: D
Let Height of the cylinder = 2r
Curved surface area of the cylinder = 2πRH
∴ Required ratio = 4πr² : 2π × r × 2r = 1 : 1
- A hemisphere and a cone have equal base. If their heights are also equal, the ratio of their curved surface will be :
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According to the question Base of hemisphere = Base of cone
i.e. radius of hemisphere = radius of cone ...(i)
and height of hemisphere = height of cone ...(ii)
We know that height of hemisphere = radius of hemisphere
⇒ height of cone = radius of hemisphere
[From (i)]
⇒ height of cone = radius of cone
[From (ii)]
Now, Curved surface area of hemisphere = 2πr²
Curved surface area of cone = πr√r² + h²
= √r² + h²(r = h)
= √2r² = πr × √2r
= √2πr²
∴ Ratio of curved surface area of hemisphere and cone
= 2πr² : √2πr² = 2 : √2 = √2 : 1Correct Option: B
According to the question Base of hemisphere = Base of cone
i.e. radius of hemisphere = radius of cone ...(i)
and height of hemisphere = height of cone ...(ii)
We know that height of hemisphere = radius of hemisphere
⇒ height of cone = radius of hemisphere
[From (i)]
⇒ height of cone = radius of cone
[From (ii)]
Now, Curved surface area of hemisphere = 2πr²
Curved surface area of cone = πr√r² + h²
= √r² + h²(r = h)
= √2r² = πr × √2r
= √2πr²
∴ Ratio of curved surface area of hemisphere and cone
= 2πr² : √2πr² = 2 : √2 = √2 : 1
