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A solid metal sphere is melted and smaller sphere of equal radii are formed. 10% of the volumeof the sphere is lost in the process. The smaller spheres have a radius, that is l/9th the largersphere. If 10 litres of paint was needed to paint the larger sphere, how many litresis need topaint all the smaller spheres?
- a)90 litre
- b)81 litres
- c)180 litres
- d)324 litres
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A solid metal sphere is melted and smaller sphere of equal radii are f...
Let radius of larger sphere and smaller sphere be R and r
►Volume of larger sphere=4 / 3 × pi × R3
►Volume of sphere left= .9 × 4 / 3 × pi × R3
►Volume of smaller sphere =4 / 3 × pi × r3
Put r = R / 9
Put r = R / 9
►Total number of spheres = .9 × 4 / 3 × pi × R3 / 4 / 3 × pi × (R / 9)3 = 729 × .9
►Surface area of larger sphere = 4 × pi × R2
►Surface area of smaller sphere = 4 × pi × R2 / 81
►Surface area of all smaller spheres = 729 ×.9 × 4 × pi × R2 / 81 = 8.1 × 4 × pi × R2
►Therefore ratio of surface area of all the smaller spheres to the big sphere = (8.1 × 4 × pi × R2) / (4 × pi × R2) = 8.1 : 1
So Answer will be 8.1 × 10 = 81
So Answer will be 8.1 × 10 = 81
Most Upvoted Answer
A solid metal sphere is melted and smaller sphere of equal radii are f...
Given information:
- A solid metal sphere is melted and smaller spheres of equal radii are formed.
- 10% of the volume of the sphere is lost in the process.
- The smaller spheres have a radius that is l/9th the larger sphere.
- 10 litres of paint was needed to paint the larger sphere.
To find:
- How many litres of paint is needed to paint all the smaller spheres?
Solution:
Let the radius of the larger sphere be r.
The volume of the larger sphere is given by V = (4/3)πr³.
Since 10% of the volume is lost in the process, the remaining volume is 90% of the original volume, which is (0.9)(4/3)πr³.
The radius of the smaller spheres is l/9th the radius of the larger sphere, which is (1/9)r.
The volume of each smaller sphere is given by V' = (4/3)π(1/9r)³ = (4/3)(1/729)πr³.
The number of smaller spheres that can be formed from the original sphere is (0.9)(4/3)πr³ / [(4/3)(1/729)πr³] = 729(0.9) = 656.1 (approx).
Therefore, the total volume of all the smaller spheres is 656.1(4/3)(1/729)πr³ = (1/729)(1752πr³) cubic units.
The ratio of the volume of the larger sphere to the total volume of all the smaller spheres is (4/3)πr³ / [(1/729)(1752πr³)] = 2430.
This means that the smaller spheres have a combined volume that is 2430 times smaller than the volume of the larger sphere.
Since 10 litres of paint was needed to paint the larger sphere, the amount of paint needed to paint each smaller sphere is 10/2430 = (1/243) litres.
The total amount of paint needed to paint all the smaller spheres is 656.1(1/243) = 81 litres.
Therefore, the answer is option (b) 81 litres.
- A solid metal sphere is melted and smaller spheres of equal radii are formed.
- 10% of the volume of the sphere is lost in the process.
- The smaller spheres have a radius that is l/9th the larger sphere.
- 10 litres of paint was needed to paint the larger sphere.
To find:
- How many litres of paint is needed to paint all the smaller spheres?
Solution:
Let the radius of the larger sphere be r.
The volume of the larger sphere is given by V = (4/3)πr³.
Since 10% of the volume is lost in the process, the remaining volume is 90% of the original volume, which is (0.9)(4/3)πr³.
The radius of the smaller spheres is l/9th the radius of the larger sphere, which is (1/9)r.
The volume of each smaller sphere is given by V' = (4/3)π(1/9r)³ = (4/3)(1/729)πr³.
The number of smaller spheres that can be formed from the original sphere is (0.9)(4/3)πr³ / [(4/3)(1/729)πr³] = 729(0.9) = 656.1 (approx).
Therefore, the total volume of all the smaller spheres is 656.1(4/3)(1/729)πr³ = (1/729)(1752πr³) cubic units.
The ratio of the volume of the larger sphere to the total volume of all the smaller spheres is (4/3)πr³ / [(1/729)(1752πr³)] = 2430.
This means that the smaller spheres have a combined volume that is 2430 times smaller than the volume of the larger sphere.
Since 10 litres of paint was needed to paint the larger sphere, the amount of paint needed to paint each smaller sphere is 10/2430 = (1/243) litres.
The total amount of paint needed to paint all the smaller spheres is 656.1(1/243) = 81 litres.
Therefore, the answer is option (b) 81 litres.
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A solid metal sphere is melted and smaller sphere of equal radii are formed. 10% of the volumeof the sphere is lost in the process. The smaller spheres have a radius, that is l/9ththe largersphere. If 10 litres of paint was needed to paint the larger sphere, how many litresis need topaint all the smaller spheres?a)90 litreb)81 litresc)180 litresd)324 litresCorrect answer is option 'B'. Can you explain this answer?
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