Sphere
Class 10th Mathematics West Bengal Board Solution
- If the length of radius of a sphere is 10.5 cm., let us write by calculating the whole…
- If the cost of making a leather ball is 431.20 at 17.50 per square cm., let us write by…
- If the length of diameter of the ball used for playing shotput in our school is 7 cm.…
- If the length of diameter of a solid sphere is 28 cm and it is completely immersed into…
- The length of radius of spherical gas balloon increases from 7 cm to 21 cm as air is…
- 127 2/7 sq cm. of sheet is required to make a hemispherical bowl. Let us write by…
- The length of radius of solid spherical ball is 2.1 cm; let us write by calculating how…
- The length of diameter of solid sphere of lead is 14 cm. If the sphere is melted, let…
- Three spheres made of copper having the lengths of 3 cm. 4 cm. and 5 cm. radii are…
- The length of diameter of base of a hemispherical tomb is 42 dm. Let us write by…
- Two hollow spheres with the lengths of diameter 21 cm and 71.5 cm respectively are…
- The curved surface of a solid metalic sphere is cut in such a way that the curved…
- On the curved surface of the axis of a globe with the length of 14 cm. radius, two…
- Let us write by calculating how many marbles with lengths of 1 cm. radius may be…
- Q15A1 The volume of a solid sphere having the radius of 2r units length isA. 32 pi r^3/3…
- Q15A2 If the ratio of the volumes of two solid spheres is 1 : 8, the ratio of their curved…
- Q15A3 The whole surface area of a solid hemisphere with length of 7 cm radius isA. 588 π…
- Q15A4 If the ratio of curved surface areas of two solid spheres is 16 : 9, the ratio of…
- Q15A5 If numerical value of curved surface area of a solid sphere is three times of its…
- Let us write whether the following statements are true of false. i. If we double the…
- Let us fill in the blanks. i. The name of solid which is composed of only one surface…
- The numerical values of volume and whole surface area of a solid hemisphere are…
- The curved surface area of a solid sphere is equal to the surface area of a solid…
- Whole surface area of a solid hemisphere is equal to the curved surface area of the…
- If curved surface area of a solid sphere is S and volume is V, let us write the value…
- If the length of radius of a sphere is increased by 50%, let us write how much…
Let Us Work Out 12
Question 1.If the length of radius of a sphere is 10.5 cm., let us write by calculating the whole surface area of the sphere.
Answer:
Given:
The radius of sphere, r = 10.5cm
Formula Used:
The whole surface area of sphere, T.S.A = 4×π×r2
The figure is attached,
⇒ T.S.A = 4×π×(10.5)2
= 1386cm2
Question 2.
If the cost of making a leather ball is 431.20 at 17.50 per square cm., let us write by calculating the length of diameter of the ball.
Answer:
Given:
Cost of making a leather ball = Rs.431.20
Cost per square cm = Rs.17.50
Formula used:
1. Cost of making a leather ball = Surface Area of ball × Cost per square cm
⇒ 431.20 = Surface area × 17.50
⇒ Surface Area = 24.64 cm2
2. surface area of sphere = 4×π×r2
⇒ 4×π×r2 = 24.64
⇒ r2 = 1.96
⇒ r = 1.4cm
So, the diameter of sphere = 2×Radius = 2.8cm
Question 3.
If the length of diameter of the ball used for playing shotput in our school is 7 cm. let us write by calculating how many cubic cm of iron is there in the ball.
Answer:
Given:
Diameter of sphere = 7cm
Formula used:
1. Diameter of sphere = 2×Radius
Radius = 7/2 = 3.5cm
2. Volume of sphere = (4/3)×π×r3
⇒ Volume of sphere = (4/3)×π×(3.5)3 = 179.67 cm3
Question 4.
If the length of diameter of a solid sphere is 28 cm and it is completely immersed into the water, let us calculate the volume of water displaced by the sphere
Answer:
Given:
Diameter of sphere = 7cm
Formula used:
1. Diameter of sphere = 2×Radius
Radius = 28/2 = 14cm
2. Volume of water displaced = Volume of sphere
(Archimedes principle)
3. Volume of sphere = 
π×r3
⇒ Volume of sphere = (
)×π×(14)3 = 11,498.67 cm3
Question 5.
The length of radius of spherical gas balloon increases from 7 cm to 21 cm as air is being pumped into it let us find the ratio of surface arcas of the balloon in two cases.
Answer:
Given:
Initial radius, r1 = 7cm
Final radius, r2 = 21cm
Formula used:
Surface area of sphere = 4×π×r2
So, Ratio of surface area 
⇒ Ratio of surface area 
Question 6.
sq cm. of sheet is required to make a hemispherical bowl. Let us write by calculating the length of diameter of the forepart of the bowl.
Answer:
Given:
Surface area of hemisphere = 127(2/7) cm2
Formula used:
1. Surface area of hemisphere = 2×π×r2
⇒ 2×π×r2 = 127(2/7)
⇒ r2 = 20.25
⇒ r = 4.5cm
2. Diameter of sphere = 2×Radius
Diameter = 9cm
Question 7.
The length of radius of solid spherical ball is 2.1 cm; let us write by calculating how much cubic cm iron is there and let us find the curved surface area of the iron ball.
Answer:
Given:
The radius of sphere = 2.1cm
Formula used:
1. Surface area of sphere = 4×π×r2
⇒ Surface area of sphere = 4×π×(2.1)2 = 55.44cm2
2. Volume of sphere = (4/3)×π×r3
⇒ Volume of sphere = (4/3)×π×(2.1)3 = 38.808 cm3
Question 8.
The length of diameter of solid sphere of lead is 14 cm. If the sphere is melted, let us write by calculating how many spheres with length of 3.5 cm. radius can be made.
Answer:
Assuming it’s the length of radius (not diameter) = 3.5cm
Given:
Diameter of big solid sphere = 14cm
Radius of small solid sphere = 3.5cm
Formulas used:
1. Diameter of sphere = 2×Radius
Radius = 14/2 = 7cm
2. Volume of big sphere = Volume of small spheres × No. of small spheres
3. Volume of sphere = (4/3)×π×r3
Volume of big sphere = (4/3)×π×(7)3 cm3
Volume of small sphere = (4/3)×π×(3.5)3 cm3
Putting the values,
⇒ (4/3)×π×(7)3 = (4/3)×π×(3.5)3 × No. of small cubes
⇒ No. of small cubes = (7)3/(3.5)3 = 8
Question 9.
Three spheres made of copper having the lengths of 3 cm. 4 cm. and 5 cm. radii are melted and a large sphere is made. Let us write by calculating the length of radius of the large sphere.
Answer:
Given:
Radius of the three small spheres are 3cm, 4cm and 5cm.
Formula used:
Let the radius of the new sphere be r.
1. Volume of 3cm radius sphere + Volume of 4cm radius sphere + Volume of 5cm radius sphere = Volume of new sphere
2. Volume of sphere = (4/3)×π×r3
⇒ (3)3+(4)3+(5)3 = r3
⇒ 27 + 64 +125 = r3
⇒ 216 = r3
⇒ r = 6
Radius of the new sphere will be 6cm.
Question 10.
The length of diameter of base of a hemispherical tomb is 42 dm. Let us write by calculating the cost of colouring the upper surface of the tomb at the rate of ` 35 per square metre.
Answer:
Given:
Diameter of base = 42dm
Formula used:
1. Diameter of hemisphere = 2×Radius
Radius = 42/2 = 21dm = 2.1m
2. Surface area of hemisphere = 2×π×r2
⇒ Surface Area = 2×π×(2.1)2 = 27.72 m2
3. Cost of coloring = Surface Area × Cost per sq m
Cost of coloring = 27.72 × 35 = Rs.970.2.
Question 11.
Two hollow spheres with the lengths of diameter 21 cm and 71.5 cm respectively are made from the sheets of the same metal. Let us calculate the volumes of sheets of metal required to make the two spheres
Answer:
Diameter of first sphere = 21 cm
Diameter of second sphere = 71.5cm
Radius of first sphere = 
Radius of second sphere = 
Volume of sheets of metal required to make sphere = Volume of sphere
Volume of Sphere=
Volume of first sphere = 
Volume of first sphere = 14539.77 cm3
Volume of second sphere=
Volume of Second Sphere = 573875.624 cm3
Question 12.
The curved surface of a solid metalic sphere is cut in such a way that the curved surface area of the new sphere is half of that previous one. Let us calculate the ratio of the volumes of the portion cut off and the remaining portion of the sphere.
Answer:
Curved surface area of initial sphere = 4πr2
Curved surface area of new sphere = (1/2) 4πr2 = 2πr2
The other half’s surface area = 2πr2
So, both the parts are hemispheres, with equal curved surface area and same radius.
Curved Surface area of new sphere is half of the curved surface area of previous one. This implies that the sphere is cut into two equal parts and hence the volumes of the two parts after cutoff will be equal too. So, the ratio will be 1:1.
Question 13.
On the curved surface of the axis of a globe with the length of 14 cm. radius, two circular holes are made each of which has the length of radius 0.7 cm. Let us calculate the area of metal sheet surrounding its curved surface.
Answer:
Given:
Radius of globe(spherical), R = 14cm
Radius of holes, r = 0.7cm
Formula used:
Surface area of sphere = 4πR2
Area of one hole(circular) = π r2
Curved surface of remaining = Curved surface area of whole sphere – Surface area of circular holes
Surface area of remaining = 4×π×R2 - 2×(π×r2)
Surface area = 4π(14)2 – 2π(0.7)2 = 2464 – 3.08 = 2460.92 cm2
Question 14.
Let us write by calculating how many marbles with lengths of 1 cm. radius may be formed by melting a solid sphere of iron having 8 cm length of radius.
Answer:
Given:
Radius of small marbles = 1cm
Radius of big sphere = 8cm
Formula used:
Volume of big sphere = Volume of small spheres × No. of marbles
We know that Volume of sphere = (4/3)×π×r3
Volume of big sphere = (4/3)×π×(8)3 cm3
Volume of small sphere = (4/3)×π×(1)3 cm3
Putting the values,
⇒ (4/3)×π×(8)3 = (4/3)×π×(1)3 × No. of marbles
⇒ No. of small marbles = (8)3/(1)3 = 512
Question 15.
The volume of a solid sphere having the radius of 2r units length is
A. 
cubic unit
B. 
cubic unit
C. 
cubic unit
D. 
cubic unit
Answer:
We know that Volume of sphere of radius r = (4/3)×π×r3
⇒ Volume of sphere of radius 2r = (4/3)×π×(2r)3 = (32/3) ×π×r3 unit3
The correct option is A.
Question 16.
If the ratio of the volumes of two solid spheres is 1 : 8, the ratio of their curved surface areas is
A. 1 : 2
B. 1 : 4
C. 1 : 8
D. 1 : 16
Answer:
Let the radius of two spheres be r1 and r2;
Ratio of volume 
⇒ 
⇒ r1:r2 = 1:2
Ratio of surface area 
The correct option is B.
Question 17.
The whole surface area of a solid hemisphere with length of 7 cm radius is
A. 588 π sq cm.
B. 392 π sq cm.
C. 147 π sq cm.
D. 98 π sq cm.
Answer:
Whole surface area of a solid hemisphere = 2×π×r2 + π×r2 = 3×π×r2
⇒ Surface Area = 3×π×(7)2 = 147 π sq. cm
The correct option is C.
Question 18.
If the ratio of curved surface areas of two solid spheres is 16 : 9, the ratio of their volumes is
A. 64 : 27
B. 4 : 3
C. 27 : 64
D. 3 : 4
Answer:
Let the radius of two spheres be r1 and r2;
Ratio of surface area
⇒ 
⇒ r1:r2 = 4:3
Ratio of volume 
The correct option is A.
Question 19.
If numerical value of curved surface area of a solid sphere is three times of its volume the length of radius is
A. 1 unit
B. 2 unit
C. 3 unit
D. 4 unit
Answer:
Curved Surface Area = 3 × Volume of sphere
⇒ 4πr2 = 3×(4/3)×π×r3
⇒ R = 1 unit
The correct option is A.
Question 20.
Let us write whether the following statements are true of false.
i. If we double the length of radius of a solid sphere, the volume of sphere will be doubled.
ii. If the ratio of curved surface areas of two hemispheres is 4 : 9, the ratio of their lengths of radii is 2 : 3.
Answer:
(i) False.
The volume of a sphere is directly proportional to the cube of the radius. So if the radius is doubled, volume becomes 8 times.
(ii) True.
Let the radius of two spheres be r1 and r2;
Ratio of surface area
⇒ 
⇒ r1:r2 = 2:3
Question 21.
Let us fill in the blanks.
i. The name of solid which is composed of only one surface is _______.
ii. The number of surfaces of a solid hemisphere is ___________.
iii. If the length of radius of a solid hemisphere is 2r units, its whole surface area is __________ π r2 sq units.
Answer:
(i) Sphere.
(ii) Curved and Flat
(iii) 
Whole surface area of a solid hemisphere of radius r = 2×π×r2 + π×r2 = 3×π×r2
Surface area of radius 2r = 3×π×(2r)2 = 12×π×r2
Question 22.
The numerical values of volume and whole surface area of a solid hemisphere are equal, let us write the length of radius of the hemisphere.
Answer:
Volume of hemisphere = Whole surface area of hemisphere
⇒ (2/3)×π×r3 = 3×π×r2
⇒ R = 9/2 = 4.5 units
Question 23.
The curved surface area of a solid sphere is equal to the surface area of a solid right circular cylinder. The lengths of both height and diameter of cylinder are 12 cm. Let us write the length of radius of the sphere.
Answer:
Given:
Height of cylinder = 12cm
Diameter of cylinder = 12cm
Formula Used:
Image source: socratic
Surface Area of cylinder = 2πrh + 2πr2
Surface Area = 2π(6×12 + 62) = 216 π
Surface Area of sphere = Surface area of cylinder
⇒ 4πr2 = 216 π
⇒ r2 = 54
⇒ r = 7.35cm
Question 24.
Whole surface area of a solid hemisphere is equal to the curved surface area of the solid sphere. Let us write the ratio of lengths of radius of hemisphere and sphere.
Answer:
Whole surface area of a solid hemisphere of radius r1 = 2×π×r12 + π×r12 = 3×π×r12
We know that surface area of sphere = 4×π×r22
3×π×r12 = 4×π×r22
⇒ 
⇒ r1:r2 = 2:√3
Question 25.
If curved surface area of a solid sphere is S and volume is V, let us write the value of 
[not putting the value of π]
Answer:
Formula used:
1. Surface area of sphere = 4×π×r2
2. Volume of sphere = (4/3)×π×r3
= 36 π
Question 26.
If the length of radius of a sphere is increased by 50%, let us write how much percent will be increased of its curved surface area.
Answer:
Let the initial radius be r.
After 50% increase, new radius r’ = r+0.5r = 1.5r
Initial Surface area = 4×π×r2
Final Surface Area = 4×π×(1.5r)2 = 9×π×r2
⇒ Percent increase = 