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.

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