Multiple Choice Questions
1. Find the volume of a sphere whose radius is 7 cm.
a)Both A and R are true and R is the correct explanation of A
b)Both A and R are true but R is not the correct explanation of A
c)A is true but R is false
d)A is false but R is true
Hint: Use 4ΟrΒ² = 154 to find r, then use V = 4/3ΟrΒ³
Solution: r = 7/2 β V = 4/3 Γ Ο Γ (7/2)Β³ = 539/3 = 178.33 cmΒ³
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Assertion Reasoning Questions
1. Assertion: 27 iron balls of radius r form a new sphere of radius 3r. Reason: Volume of sphere is proportional to cube of radius.
a)7Ο
b)7R/C
c)173.23 cmΒ³
d)(83/21)Β³
Hint: V β rΒ³ β 27rΒ³ = (rβ²)Β³ β rβ² = 3r
Solution: Volume preserved β rβ² = β(27rΒ³) = 3r
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True False Questions
1. Volume of iron used in hemispherical tank with inner radius 1 m and 1 cm thick is 0.064 mΒ³.
Hint: Correct value is 0.06348 mΒ³
Solution: Check V = 2/3Ο(RΒ³ β rΒ³) with R = 1.01, r = 1
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2. A capsule of diameter 3.5 mm contains approximately 22.46 mmΒ³ of medicine.
Hint: Use V = 4/3ΟrΒ³, r = 1.75 mm
Solution: 4/3 Γ 22/7 Γ 1.75Β³ β 22.46 mmΒ³
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Fill In The Blanks
1. Ratio of surface areas S and Sβ² for original and new sphere (r and 3r) = ___
Hint: S β rΒ² β (r)Β² : (3r)Β² = 1 : 9
Solution: 4ΟrΒ² : 4Ο(3r)Β² = 1 : 9
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2. Volume of air inside hemispherical dome of radius 6.3 m = ___ mΒ³.
Hint: Use V = 2/3ΟrΒ³
Solution: 2/3 Γ 22/7 Γ 6.3Β³ = 523.9 mΒ³
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Descriptive Questions
1. **Case::**
In a research laboratory, a scientist is studying the properties of spherical particles. One such particle has a surface area of 154 cmΒ². The scientist needs to determine the exact volume of the particle to understand its density and how it interacts in chemical reactions. To find the volume, they must first calculate the radius using the surface area formula and then substitute that value into the volume formula.
(i) What is the radius of the sphere?
(ii) What is its volume?
(iii) Why is it necessary to find radius before volume?
Hint: First apply SA = 4ΟrΒ², then use V = (4/3)ΟrΒ³
Solution: (i)
- SA = 4ΟrΒ² = 154 β rΒ² = 154 Γ· 4Ο = 49 β r = 7/2 = 3.5 cm
(ii)
- V = (4/3)Ο(3.5)Β³ = 179.66 cmΒ³
(iii)
- Volume requires radius β r found from surface area
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2. **Case::**
A team of engineers is tasked with constructing a hemispherical water tank using iron sheets. The inner radius of the tank is 1 meter, and the iron sheet used to make it is 1 cm thick. The engineers need to determine how much iron is required to manufacture this tank. To do this, they calculate the volume of the outer and inner hemispheres and subtract them to find the volume of the iron shell. This calculation ensures that material procurement is precise and prevents any overestimation or wastage.
(i) What is the volume of iron used in constructing the tank?
(ii) What are the inner and outer radii in meters?
(iii) Why is volume of iron calculated using 2/3Ο(RΒ³ β rΒ³)?
Hint: Subtract volumes of outer and inner hemispheres
Solution: (i)
- R = 1.01 m, r = 1 m
- V = (2/3)Ο(RΒ³ β rΒ³) = (2/3) Γ (22/7) Γ (1.030301 β 1) = 0.06348 mΒ³
(ii)
- Inner radius = 1 m, Outer = 1 + 0.01 = 1.01 m
(iii)
- Hemispherical shell β V = (2/3)Ο(RΒ³ β rΒ³)
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