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?
To find out the area of the field Zoe can graze, we need to consider the shape of her grazing area. As Zoe is tied to one corner of the rectangular barn, her grazing area will be a quarter circle with the barn corner as its center.
Let's calculate the area of the grazing area for each length of Zoe's rope:
1. When the rope is 10 meters long:
- The radius of the quarter circle will be equal to the length of the rope, so r = 10 meters.
- To find the area of the quarter circle, we use the formula: A = (π * r^2) / 4.
- Substituting the value, we get A = (π * 10^2) / 4 = 25π square meters.
- Zoe can graze an area of approximately 78.54 square meters.
2. When the rope is 20 meters long:
- The radius of the quarter circle will be 20 meters.
- Using the same formula, A = (π * 20^2) / 4 = 100π square meters.
- Zoe can graze an area of approximately 314.16 square meters.
3. When the rope is 30 meters long:
- The radius of the quarter circle will be 30 meters.
- Using the formula again, A = (π * 30^2) / 4 = 225π square meters.
- Zoe can graze an area of approximately 706.86 square meters.
Therefore, Zoe can graze an area of approximately 78.54 square meters with a rope of length 10 meters, 314.16 square meters with a rope of length 20 meters, and 706.86 square meters with a rope of length 30 meters.
To determine the area in which Zoe the goat can graze, we need to consider the length of her rope and the shape of the available area.
For a 10-meter long rope:
1. Start by visualizing the barn in the field and Zoe tied to one corner with a 10-meter rope.
2. The goat can move in an arc, as restricted by the rope length, centered at the corner to which she is tied.
3. This results in a quarter-circle of radius 10 meters.
4. To find the area, calculate the area of the quarter-circle using the formula: A = (π * r^2) / 4, where r is the radius.
5. Substituting the given radius of 10 meters into the formula, the area is: A = (π * 10^2) / 4 = 78.54 square meters.
For a 20-meter long rope:
1. Similarly, visualize Zoe tied to one corner with a 20-meter rope.
2. This time, the goat's grazing area will form a semi-circle.
3. To find the area, calculate the area of the semi-circle using the formula: A = (π * r^2) / 2, where r is the radius.
4. Substituting the given radius of 20 meters into the formula: A = (π * 20^2) / 2 = 628.32 square meters.
For a 30-meter long rope:
1. Finally, visualize Zoe tied to one corner with a 30-meter rope.
2. In this case, the goat has more freedom to roam within the field.
3. Since the rope length exceeds the diagonal length of the barn (using the Pythagorean theorem: √(15^2 + 25^2) ≈ 29.15 meters), Zoe can reach any point within the rectangular field.
4. Thus, the grazing area will be the same as the total area of the rectangular field, which is: A = length * width = 15 * 25 = 375 square meters.
So, depending on the length of the rope, Zoe can graze approximately 78.54 square meters (with a 10-meter rope), 628.32 square meters (with a 20-meter rope), or 375 square meters (with a 30-meter rope).