Zoe the goat is tied by a rope to one corner of a 15 meter-by-25 meter rectangular barn in the middle of a large grassy field. Over what area of the field can Zoe graze if the rope is 10 meters long? 20 meters long? 30 meters long?
Where did you get the 5?
a 10m rope allows grazing over 3/4 of a 10m circle
3/4 * π * 100 = 75π
a 20m rope allows grazing over 3/4 of a 20m circle + 1/4 of a 5m circle
3/4 * π * 400 + 1/4 * π * 25 = 306.25 π
a 30m rope allows
3/4 of a 30m circle
+ 1/4 of a 15m circle
+ 1/4 of a 5m circle
To find the area that Zoe can graze, we need to consider the shape of the region she can reach, which is a circular shape centered at the corner of the barn. The radius of the circular region is determined by the length of the rope.
Let's calculate the areas for different rope lengths:
1. Rope length: 10 meters
The radius of the circular region is 10 meters.
The area of a circle is given by the formula: A = π * r^2.
Substituting the values, we have: A = π * 10^2 = 100π square meters.
Therefore, Zoe can graze an area of 100π square meters.
2. Rope length: 20 meters
The radius of the circular region is 20 meters.
The area of a circle is given by the formula: A = π * r^2.
Substituting the values, we have: A = π * 20^2 = 400π square meters.
Therefore, Zoe can graze an area of 400π square meters.
3. Rope length: 30 meters
The radius of the circular region is 30 meters.
The area of a circle is given by the formula: A = π * r^2.
Substituting the values, we have: A = π * 30^2 = 900π square meters.
Therefore, Zoe can graze an area of 900π square meters.
So, depending on the length of the rope, Zoe can graze an area of 100π square meters, 400π square meters, or 900π square meters.
To find the area over which Zoe can graze, we need to determine the shape that the rope boundary creates on each side. Let's break it down for different rope lengths:
1. Rope length of 10 meters:
- When Zoe is tied with a 10-meter rope, the shape created will be a quarter circle since the rope stretches out from the corner of the barn.
- The radius of the quarter-circle is the length of the rope, which is 10 meters.
- The area of a quarter circle is given by the formula A = πr²/4, where A is the area and r is the radius.
- Plugging in the values, we get A = (π * 10²) / 4 = 25π square meters.
2. Rope length of 20 meters:
- In this case, the rope stretches further and forms a semicircle rather than a quarter circle.
- The radius of the semicircle is the length of the rope, which is 20 meters.
- The area of a semicircle is given by the formula A = πr²/2, where A is the area and r is the radius.
- Substituting the values, we get A = (π * 20²) / 2 = 200π square meters.
3. Rope length of 30 meters:
- With a 30-meter rope, the rope boundary extends even further and forms a half circle.
- The radius of the half circle is the length of the rope, which is 30 meters.
- The area of a half circle is given by the formula A = πr², where A is the area and r is the radius.
- Substituting the values, we get A = π * (30²) = 900π square meters.
So, Zoe can graze over an area of 25π square meters with a 10-meter rope, 200π square meters with a 20-meter rope, and 900π square meters with a 30-meter rope.