Question

A man running round a race course notes that the sum of the distance of two flag - posts from him is always 10 meters and distance between the flag - posts is 8 meters. The area of the path he encloses in square meters is (1) \(15 \pi\) (2) \(12 \pi\) (3) \(18 \pi\) (4) \(8 \pi\)

Short answer

The area of the path enclosed is \(15 \pi\) square meters.

Step-by-step solution

Visualize and Frame the Problem

Imagine the racecourse as an ellipse with the man running along its perimeter. The two flag-posts are the foci of the ellipse. The given condition is that the sum of the distances from any point on the ellipse to the two foci (flag-posts) is constant and equal to 10 meters.

Understand Ellipse Properties

In an ellipse, the sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis. Therefore, the length of the major axis is 10 meters.

Determine Focal Distance

The distance between the two foci (flag-posts) is given as 8 meters. In an ellipse, this distance is equal to 2c, where c is the distance from the center to each focus. Thus, 2c = 8 meters, so c = 4 meters.

Calculate Semi-Major and Semi-Minor Axes

The length of the major axis (2a) is 10 meters, so the semi-major axis (a) is 5 meters. Use the relationship in ellipses: \(a^2 = b^2 + c^2\). Plug in the values: \(5^2 = b^2 + 4^2\), thus \(25 = b^2 + 16\). Solving for b, we get \(b^2 = 9\) or \(b = 3\) meters.

Compute the Area of the Ellipse

The area of an ellipse is given by \(A = \pi a b\). Using a = 5 meters and b = 3 meters, the area becomes \(A = \pi \times 5 \times 3 = 15\pi\) square meters.

Key concepts

Ellipse Geometry

An ellipse is a fascinating geometric shape, looking like an elongated circle. When you're dealing with ellipses, there are two focal points (or foci). Each point on the ellipse maintains a constant total distance to these two foci. In the given exercise, the sum of these distances is 10 meters, which is key to understanding the shape and size of the ellipse.
For more clarity, think of the racecourse the man runs as being an ellipse. The two flag-posts are the foci. The sum of the distances from the man’s position to each flag-post is 10 meters. This characteristic property of ellipses helps us define and calculate other essential measurements such as the major axis.

Distance Formulas

Distance formulas are crucial for geometric calculations. For the ellipse, there are two important distances - the major axis and the distance between the foci.
The length of the major axis is the maximum distance across the ellipse and in this exercise, it is 10 meters since this is the constant sum of the distances from any point on the ellipse to the foci.
To find the focal distance (distance between the two foci), we use the property that this distance equals 2c. Given that this distance is 8 meters, we solve for c (4 meters).
Another important equation is the relationship between the semi-major axis (a), semi-minor axis (b), and the distance to the foci (c): \( a^2 = b^2 + c^2 \). This helps us to find b when we know a and c.

Geometric Area Calculation

Calculating the area of an ellipse combines understanding its geometry and using specific formulas. The area (A) of an ellipse is given by the formula \( A = \pi ab \), where a is the semi-major axis and b is the semi-minor axis.
We already determined that the semi-major axis (a) is 5 meters and, using the relationship \( a^2 = b^2 + c^2 \), we found that the semi-minor axis (b) is 3 meters. Plugging these values into the area formula gives:
\[ A = \pi \times 5 \times 3 = 15\pi \text{ square meters} \]
This calculation provides the area enclosed by the man's path on the racecourse, confirming that the correct answer is \( 15\pi \) square meters.