Understanding the Horse Grazing Problem

The problem describes a horse tied to a pole with a rope. As the horse grazes, the rope keeps it a fixed distance from the pole. When the horse moves from one point to another while the rope is taut, it follows the path of an arc. The area the horse grazes in this scenario forms a sector of a circle.

In this specific problem:

• The pole is the center of the circle.
• The length of the rope is the radius of the circle.
• The path traced by the horse from point A to point B is an arc of the circle.
• The area grazed between the pole, the two points A and B, and the arc connecting them is the area of a sector.

Identifying Given Information for Sector Area

We are given the following information:

• Length of the rope (radius of the sector), \(r = 6\) m.
• The angle swept by the horse's movement (central angle of the sector), \( \theta = 70^\circ \).

We need to find the area of this sector.

Calculating the Area of the Sector

The formula for the area of a sector of a circle with radius \(r\) and central angle \( \theta \) (in degrees) is:

\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]

Now, we substitute the given values into the formula:

\[ \text{Area} = \frac{70^\circ}{360^\circ} \times \pi (6 \text{ m})^2 \]

Simplify the fraction and the radius term:

\[ \text{Area} = \frac{70}{360} \times \pi \times 36 \text{ m}^2 \]

\[ \text{Area} = \frac{7}{36} \times \pi \times 36 \text{ m}^2 \]

We can cancel out the 36 in the denominator and numerator:

\[ \text{Area} = 7 \times \pi \text{ m}^2 \]

To get a numerical value, we use the approximate value of \( \pi \approx \frac{22}{7} \), which is commonly used in such problems to yield simple results:

\[ \text{Area} \approx 7 \times \frac{22}{7} \text{ m}^2 \]

\[ \text{Area} \approx 22 \text{ m}^2 \]

So, the area of the sector grazed by the horse is approximately 22 m\(^2\).

Area of Sector Calculation Summary

Based on our calculation, the area grazed by the horse is 22 square meters.

Revision Table: Circle Geometry Formulas

Additional Information on Sector and Arc

A sector is essentially a slice of a circle. It is defined by two radii and the arc between them. The size of the sector is determined by the angle between the two radii, called the central angle.

• The area of the sector is a fraction of the total area of the circle, determined by the ratio of the central angle to the total angle in a circle (360 degrees).
• The arc length is the length of the curved boundary of the sector. It is also a fraction of the total circumference of the circle, determined by the same angle ratio.
• Understanding sectors and arcs is crucial in problems involving parts of circles, such as calculating areas for curved regions or distances along curved paths. This horse grazing problem is a classic example of applying the sector area concept.