Q: If the goat is tied to a corner and the rope length is $$r = 100$$, what is the length of the arc that forms the boundary of the grazed area?

$$s = \frac{1}{4} (2 \pi \times 100) = 50 \pi$$

Question 1

A goat is tied by a rope of length $$r$$ meters to the corner of a square field with sides of length $$100$$ meters, as shown in the diagram. If the goat is able to reach exactly half the grass in the field, what is the value of $$r$$?

Accepted Answer

$$r = 100 \sqrt{2/\pi}$$

Question 2

Given the same scenario, what is the area accessible to the goat if the rope length is $$r = 100$$ meters?

Accepted Answer

$$A = \frac{1}{4} \pi (100)^2$

Question 3

If the goat's rope is long enough to reach the entire field, what is the minimum value of $$r$$?

Accepted Answer

$$r = 100 \sqrt{2}$$

Question 4

If the goat is tied to the center of the square field with side $$100 m $$and the rope length is $$50 m$$, what fraction of the field can the goat graze?

Accepted Answer

$$\frac{\pi (50)^2$}{100^2$}$$

Question 5

If the goat is tied to a corner and the rope length is $$r$$, what is the area the goat can graze, assuming $$r \u003c 100$$?

Accepted Answer

$$A = \frac{1}{4} \pi r^2$

Question 6

If the goat is tied to the midpoint of one side of the square and the rope length is $$r = 100$$, what is the area the goat can graze?

Accepted Answer

$$A = \frac{1}{2} \pi (100)^2$

Question 7

If the goat is tied to a corner and the rope length is $$r = 100$$, what is the length of the arc that forms the boundary of the grazed area?

Accepted Answer

$$s = \frac{1}{4} (2 \pi 	imes 100) = 50 \pi$$

Question 8

If the goat is tied to a corner and the rope length is $$r$$, what is the perimeter of the grazed region (excluding the field boundaries)?

Accepted Answer

$$P = \frac{1}{4} (2 \pi r)$$

Question 9

If the goat is tied to a corner and the rope length is $$r = 100$$, what is the maximum straight-line distance the goat can reach from the corner?

Accepted Answer

$$100$$ meters

Question 10

If the goat is tied to a corner and the rope length is $$r = 100$$, what is the area of the field that the goat cannot reach?

Accepted Answer

$$100^2$$ - \frac{1}{4} \pi $$(100)^2$$

Calculus Chapter 1: Functions, Radian Measure, and Trigonometric Functions

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