Volume and Displacement Calculation: A Right Circular Cylinder and Sphere

In this article, we will explore the mathematical calculations involved in determining the volume of a right circular cylinder with a given radius and height, and the volume and displacement of a sphere when it is dipped into the cylinder. This problem is a fundamental example in geometry and fluid mechanics, illustrating the principles of volume, displacement, and overflow.

Volume Calculation of the Right Circular Cylinder

The volume of a right circular cylinder can be calculated using the formula:

V πr2h

Given that the radius of the base of the cylinder is 20 meters and the height is 10 meters, we can calculate the volume as follows:

V π × (202) × 10
V 22/7 × 20 × 20 × 10

Let's denote the volume of the cylinder as A:

A 2210 cubic meters

Volume Calculation of the Sphere

The volume of a sphere can be calculated using the formula:

V1 4/3πr3

Given that the radius of the sphere is 5 meters, we can calculate the volume as follows:

V1 4/3 × π × (53)
V1 4/3 × 22/7 × 5 × 5 × 5

Let's denote the volume of the sphere as A1:

A1 4/3 × 2210/7
A1 1100/7 ≈ 157.14 cubic meters

Displacement and Overflow Calculation

When a sphere is dipped into the cylinder, the volume of water displaced by the sphere is equal to the volume of the sphere itself. This can be calculated as the ratio of the volume of the sphere to the volume of the cylinder:

Volume of water displaced / Volume of cylinder A1 / A
A1 / A (4/3 × 2210/7) / 2210

Simplifying further:

A1 / A (4/3 × 2210/7) / (2210/1)
A1 / A (4/3 × 2210/7) × (1/2210)

A1 / A 4/3 × 1/7
A1 / A 4/21

Since the volume of the sphere is less than the volume of the cylinder, only a fraction of the sphere will displace the water. To find out what fraction of the water will overflow, we need to consider the total volume of the cylinder and the volume of the sphere:

Overflow fraction Volume of sphere / Volume of cylinder
Overflow fraction A1 / A
Overflow fraction 157.14 / 2210
Overflow fraction ≈ 0.0712

Therefore, approximately 7.12% of the water will overflow when the sphere is dipped into the cylinder.

Answer: The fraction of water that will overflow is approximately 1/14.1 (or 0.0712) of the total water in the cylinder.