6 tennis balls of diameter 62 mm are placed in a cylindrical tube as shown in the figure. Find the volume of the internal unfilled space in the tube and express this as the percentage of the volume of the tube.

Chapter/Topic: Comparing Quantities

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13.33%

23.33%

33.33%

43.33%

answer is C.

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Detailed Solution

Each tennis ball is a sphere and the tube is a cylinder.
seo

We will calculate the radius of the sphere by substituting 62 for diameter in the formula

Radius = \frac{Diameter}{2}

r = \frac{62}{2} \Rightarrow r = 31 mm

The radius of the sphere is 31 mm. We will calculate the volume of the sphere by substituting 31 for r in the formula, VS=

\frac{4}{3} π r^{3}

. Therefore, we get
VS=

\frac{4}{3} π {(31)}^{3}

Applying the exponent on the term, we get
VS=

\frac{4}{3} π (29791)

Simplifying the above expression, we get
⇒VS=124788.279mm3
We will calculate the volume occupied by 6 spheres:
6VS = 6×124788.249=748729.494mm3
We will calculate the volume of the cylinder.
We can see in the figure that the radius of the cylinder is the same as the radius of the sphere and the height of the cylinder is 6 times the diameter of the sphere.
r = 31mm
h = 6×62 ⇒ h = 372mm
We will substitute 31 for r and 372 for h in the formula VC = πr2h and calculate the volume of the cylinder:
VC = π(31)2⋅372
⇒VC = 1123094.24mm3
The volume of internal unoccupied space will be the difference of the volume of the cylinder and the volume of 6 spheres.
V=VC−6VS
Substituting the values of volumes, we get
⇒V=1123094.24−748729.494
Subtracting the terms, we get
⇒V=374364.746mm3
We will express this volume as a percentage of the volume of the tube. We will substitute 374364.746 for the given volume and 1123094.24 for the total volume in the formula,

\frac{given value}{total value} \times 100

. Therefore, we get

\frac{374364.746}{1123094.24} \times 100 = 33.33 %

∴ The volume of the internal unfilled space is 374364.746mm3 and it is 33.33% of the total volume of the cylinder.
Hence, Option (3) is the correct answer.

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