Austin found that the volume of a tennis ball is 36 cubic cm. He has a cylindrical container with a radius of 6 cm and a
height of 20 cm. Will Austin be able to fit three tennis balls into the container? How do you know?

Question

08-02-2022
Mathematics

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Austin found that the volume of a tennis ball is 36 cubic cm. He has a cylindrical container with a radius of 6 cm and a
height of 20 cm. Will Austin be able to fit three tennis balls into the container? How do you know?

Sagot :

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Sqdancefanidn Sqdancefanidn · Answer 1 · 2022-02-09T22:15:03-05:00

Answer:

Yes, 3 balls will fit; the ball radius is smaller than the container radius, and the height is more than 4 times the ball diameter.

Step-by-step explanation:

We can compare the radius of a tennis ball to the radius and height of the cylinder to see if the balls will fit.

The volume of a sphere is given by ...

V = 4/3πr³

Solving for radius, we find ...

r = ∛(3V/(4π))

For the given ball volume of 36 cm³, the ball radius is found to be ...

r = ∛(3·36/(4π)) = ∛(27/π) ≈ 2.05 . . . cm

The radius of the container is 6 cm, almost 3 times the radius of a ball, so we know 3 tennis balls will fit in one layer in the container. Actually, the attached figure shows that 5 tennis balls will fit in one layer in the container.

The height of the container, 20 cm, is almost 10 times the radius of the tennis ball, 5 times the diameter, so we can get at least 5×5 = 25 balls in the container.

_____

Additional comment

The height of the container is 20/(2·2.05) ≈ 4.88 times the diameter of the tennis ball. When spheres are packed, each layer after the first takes additional height of at most √3/2 ≈ 0.87 times the diameter of the sphere. That means we can pack 5 layers of tennis balls in the container and have room left over.

Sagot :

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