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In a rectangular box of dimensions 20 centimetres by 20 centimetres by 100 centimetres, identical spherical balls are arranged in layers such that each layer has exactly 4 balls. The top view of this arrangement is shown in the figure above. What is the approximate percentage of space left empty inside the box, if the box contains the maximum possible number of layers of such spherical balls? (The volume of a sphere is  , where r is the radius of the sphere)
  • a)
    33%
  • b)
    40%
  • c)
    50%
  • d)
    60%
  • e)
    67%
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
In a rectangular box of dimensions 20 centimetres by 20 centimetres by...
Given
  • Dimension of rectangular box = 20 by 20 by 100 centimeters
  • Number of spherical balls in each layer = 4
    • Volume of a sphere =
    • , where r is the radius of the sphere
To Find: Approximate percentage of empty space left inside the box?
Approach
2. Volume of box
a. As we know the dimensions of the box, we can find the volume of the box
    3. Volume of total spheres in the box
  • As all the spheres in the box are identical, they will have the same volume. So, we can write
    1. (Volume of total spheres in the box) = (Volume of 1 sphere * Number of spheres in the box).
  • As we can see that the {length of the box} = [2 * diameter of the sphere] = [4 * radius of the sphere]
    1. Since, we know the length of the box, we can find the radius of a spherical ball
    2. Thus, we can find the volume of a sphere, however we also need to find the number of spheres in the box
       
  • Number of spheres in the box
    1. Each layer of the spherical balls will have a height equal to the diameter of a spherical ball.
    2. Since the balls in a layer are aligned along the dimensions 20 by 20, the number of layers of spherical balls will depend on the height of the box, i.e. 100 centimeters
      1. We can write this equation because we are given that the number of layers in the box is the maximum possible. Had this relation not been given we could not have used the height of the box in the numerator of the equation.​
Working Out
  1. [ Volume of box = 20 * 20 * 100 (cm)3]
  2. Volume of spheres in the box
    1. Let’s assume the radius of a spherical ball to be r centimeters
      1. So, 4r = 20 i.e. r = 5 centimeters
      2. Thus, volume of a sphere = 
      3. ​​
      4. [ Total volume of spheres in the box
  3. . Percentage of empty space left inside =
Answer: C
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In a rectangular box of dimensions 20 centimetres by 20 centimetres by 100 centimetres, identical spherical balls are arranged in layers such that each layer has exactly 4 balls. The top view of this arrangement is shown in the figure above. What is the approximate percentage of space left empty inside the box, if the box contains the maximum possible number of layers of such spherical balls? (The volume of a sphere is , where r is the radius of the sphere)a)33%b)40%c)50%d)60%e)67%Correct answer is option 'C'. Can you explain this answer?
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