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# Solved: Grazing goat problems. Consider the following ISBN: 9780321570567 2

## Solution for problem 48E Chapter 10.3

Calculus: Early Transcendentals | 1st Edition

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Problem 48E

Grazing goat problems. Consider the following sequence of problems related to grazing goats tied to a rope. (See The Guided Projects for more 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 ≤ a ≤ 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:

Solution 48E

Step 1:

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 ≤ a ≤ 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 2:

This can be drawn as To find the area of the grassy region that the goat can graze Step 3:

We have to find the area of the shaded region .

Let us consider the distance Therefore by pythagoras theorem we get Now lets us consider the triangle RPQ and RQP

Let us take and Thus we get And Step 4:

Now to find the area of the shaded region we will have to add the area of the segments S1 and S2.

Area of a segment can be found by adding the triangle and the sector.

Therefore area of the shaded region is Step 5 of 7

Step 6 of 7

##### ISBN: 9780321570567

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