85–87. Grazing goat problems Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.)


A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a rope of length 0≤a ≤ π (see figure). What is the area of the region (outside the corral) that the goat can reach?

Verified step by step guidance

1. Identify the problem setup: A goat is tied outside a circular corral of radius 1 with a rope of length \(a\), where \(0 \leq a \leq \pi\). The goat can move along the fence and also away from it, but only up to the length of the rope.
2. Visualize the reachable area: The goat's reachable region outside the corral is formed by the path along the fence (an arc of length \(a\)) plus the area the goat can graze around the point where the rope ends, which forms a circular segment outside the corral.
3. Express the length of the arc along the fence: Since the corral has radius 1, the length of the arc the goat can move along is \(a\), which corresponds to an angle \(\theta = a\) radians on the unit circle.
4. Calculate the area accessible to the goat: The total reachable area outside the corral is the sum of the area of the circular segment formed by the rope length \(a\) and the area of the circular sector of radius \(a\) that the goat can graze beyond the fence. This can be expressed as the area of a circular sector of radius \(a\) minus the area of the triangular portion under the chord, plus the area of the arc segment along the fence.
5. Formulate the area mathematically: Use the formula for the area of a circular sector \(\frac{1}{2} a^2 \theta\) and the area of the triangular portion \(\frac{1}{2} a^2 \sin(\theta)\) to write the grazing area as \(\frac{1}{2} a^2 (\theta - \sin(\theta))\), where \(\theta = a\). Combine this with the arc length \(a\) along the fence to get the total reachable area.

Key Concepts

Geometry of Circles and Arcs

Understanding the properties of circles, including radius, circumference, and arc length, is essential. The goat is tied to a point on the circular fence, and the rope length determines the arc along which the goat can move. Calculating the reachable area involves analyzing segments and arcs of the circle.

Area Calculation Using Integration

To find the area accessible to the goat, one must set up and evaluate integrals that represent the region outside the corral but within the rope's reach. This involves integrating over circular segments or sectors, considering the constraints imposed by the fence and rope length.

Parametric and Polar Coordinates

Using parametric or polar coordinates simplifies the description of points on and around the circular corral. These coordinate systems help express the boundary conditions and limits of integration more naturally when dealing with circular shapes and arcs.