Problem 60 - Calculus Early Transcendentals

Question

Consider the following sequence of problems related to grazing goats tied to a rope. (See the Guided Project Grazing goat problems.) A circular concrete slab of unit radius is surrounded by grass. A goat is tied to the edge of the slab with a rope of length \(0 \leq a \leq 2\) (see figure). What is the area of the grassy region that the goat can graze? Note that the rope can extend over the concrete slab. Check your answer with the special cases \(a=0\) and \(a=2\).

Step-by-Step Solution

Answer

Answer: The area of the grassy region the goat can graze depends on the length of the rope (a). For 0 < a ≤ 1, the area is A = 1/2 * π * a^2. For 1 < a ≤ 2, the area is A = 1/2 * π * a^2 + 1/2 * π * (a-1)^2.

Step 1: Case 1: a = 0

When the length of the rope is 0, the goat cannot move from the point where it is tied. Thus, it cannot graze any grass. Hence, the area of the grassy region the goat can graze is 0.

Step 2: Case 2: 0 < a ≤ 1

In this case, the rope is long enough for the goat to graze grass, but not long enough to reach the center of the concrete slab. The goat will be able to graze a circular sector of grass with a radius equal to the length of the rope (a). Let's call the angle of the circular sector θ. The area of the circular sector A can be calculated as A = 1/2 * θ * a^2. We need to calculate the value of the angle θ. Since the goat is tied to the edge of the slab, the angle will be 180 degrees or π radians. So the area of the grassy region is A = 1/2 * π * a^2.

Step 3: Case 3: 1 < a ≤ 2

In this case, the length of the rope extends beyond the center of the concrete slab. We will now have two circular sectors - one outside the slab and another one containing the covered concrete slab. First, let's calculate the area of the grassy sector outside the circular slab. The area can be calculated using the same formula as before: A = 1/2 * θ * a^2, where θ is 180 degrees or π radians. The area of the circular sector containing the concrete slab can be calculated in a similar manner, considering that the difference between the original circle the goat can walk and the circle within the slab. The length of the rope in this case should be (a-1), and the angle remains the same, θ = π. So, the area of the grassy region inside the slab is A = 1/2 * π * (a-1)^2. The total area of the grassy region that the goat can graze is the sum of both sectors: A = 1/2 * π * a^2 + 1/2 * π * (a-1)^2. To check the results for the special cases: - When a = 0: Area = 0 - When a = 2: Area = 1/2 * π * 2^2 + 1/2 * π * (2 - 1)^2 = 3 * π (Considering that the total area is π when the rope extends to the center of the slab)

Key Concepts

Circular Sector Area

A circular sector is essentially a 'slice' of a circle, much like a slice of pie. When we're discussing the area that a goat can graze, we are looking at circular sectors of grass that the goat can reach depending on the length of its rope. If the rope forms a radius of a circle, then the circular sector's area is determined by the rope length and the angle it creates.

The formula used to find the area of a circular sector is \( A = \frac{1}{2} \theta r^2 \), where \( \theta \) is the angle in radians and \( r \) is the radius (or the length of the rope in this context). This formula helps calculate the gymnastic area where our goat can play with grass, while tied to the concrete slab. The angle for the problems regarding the goat is often \( \pi \) or 180 degrees, as the goat can stretch around to half the circle's edge.

Concrete Slab

The concrete slab in this problem acts as a barrier or a fixed boundary point where the goat is tied. With a unit radius, the slab is a perfect circle around which the goat can graze depending on rope length.

Here’s how to think about it:

• If the rope is short (e.g., for \( 0 < a \leq 1 \)), the concrete does not affect the grazing area, as it fully stands outside the concrete slab.
• For longer ropes \( (1 < a \leq 2) \), the goat grazes both over the grass and some area of the slab.

Understanding how the concrete slab and grazable area interact illustrates how the area available to the goat is sometimes limited to just what's outside the slab, and sometimes it also considers a portion that spans the slab.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. In problems like these, trigonometry helps us understand how angles play into calculating areas, like those of circular sectors.

When we use trigonometry in grazing goat problems, it often involves converting degrees to radians. Radians are more mathematically versatile in such calculations, especially when using formulas for circular sectors. Remember, 180 degrees equals \( \pi \) radians, which is crucial for these calculations because the grazing area is a half-circle arc setup.

• Use \( \pi \) as an angle in calculations to aid in solving problems quickly.
• Relate it to circular sectors via basic trigonometric identities.

These basics are handy when analyzing the simplest and most complex scenarios in goat grazing problems.

Special Cases Analysis

Through special cases, we analyze unique scenarios to predict outcomes or simplify problems. With the grazing goat, different rope lengths provide insightful specifics. For instance, special values of \( a \) such as 0 and 2 yield simple answers to help conceptualize complex situations.

• When \( a = 0 \), the goat cannot graze any grass, with the area simply being 0.
• When \( a = 2 \), the calculation becomes a bit more elaborate, as the rope allows the goat to cover the entire circle.

Examining special cases gives invaluable practice and insight since they represent 'boundaries' where potential equations shift or where barely different situations occur—helping you grasp continuum within this mixed geometry-farming problem.