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 the area over which Zoe can graze, we need to consider the shape of a circle with the rope length as the radius. Let's calculate the areas for different rope lengths:
1. Rope length = 10 meters:
The rope length is smaller than the shorter side of the rectangular barn (15 meters). Therefore, Zoe cannot reach the entire area of the field. To calculate the area, we need to find the area of the circular region.
The area of a circle is given by the formula: A = π * r^2
Substituting the value: A = π * 10^2 = 100π square meters
2. Rope length = 20 meters:
The rope length is larger than the shorter side of the rectangular barn (15 meters). However, it is still smaller than the longer side of the barn (25 meters). Consequently, Zoe can cover the entire width but not the length of the barn.
To visualize the grazing area, we have one semicircle (reaching the longer side), while the shorter side is covered fully.
The semicircle's area is given by: A(semi) = (π * r^2) / 2 = π * 20^2 / 2 = 400π / 2 = 200π square meters
The area of the rectangular part is the multiplication of its sides: A(rectangle) = 15 * 25 = 375 square meters
Adding these two parts, Zoe can graze over a total area of 200π + 375 square meters.
3. Rope length = 30 meters:
Now, the rope length is larger than both sides of the rectangular barn. Thus, Zoe can cover the entire rectangular area and some additional area beyond it.
The area of a rectangle is: A(rectangle) = 15 * 25 = 375 square meters
The additional area covered by the circular region (beyond the rectangle) is a sector of a circle.
The area of a sector is given by: A(sector) = (∠θ/360°) * π * r^2
Here, θ is the central angle corresponding to the portion of the circle that is beyond the rectangle. We can calculate it using trigonometry.
sin(θ) = (15/30)
θ = arcsin(15/30)
A(sector) = ((arcsin(15/30))/360°) * π * 30^2 = (0.5π - arcsin(15/30)/360°) * 900π square meters
So, for a rope length of 10 meters, Zoe can graze over an area of 100π square meters. For a rope length of 20 meters, Zoe can graze over an area of 200π + 375 square meters. And for a rope length of 30 meters, Zoe can graze over an area of (0.5π - arcsin(15/30)/360°) * 900π + 375 square meters.
To solve this problem, we need to understand the concept of a circle and how it relates to the grazing area of a goat tied with a rope.
The grazing area of Zoe the goat is the area she can reach while tied to her rope. This area forms a circle with Zoe's position as the center and the rope as the radius. The radius of the circle is equal to the length of Zoe's rope.
To find the area of the circle, we can use the formula for the area of a circle, which is:
Area = π * (radius)^2
Now let's calculate the grazing area for the different lengths of the rope:
1. Rope length = 10 meters:
Radius = 10 meters
Area = π * (10)^2 = 100π square meters
2. Rope length = 20 meters:
Radius = 20 meters
Area = π * (20)^2 = 400π square meters
3. Rope length = 30 meters:
Radius = 30 meters
Area = π * (30)^2 = 900π square meters
So, Zoe can graze over an area of 100π square meters with a 10-meter rope, 400π square meters with a 20-meter rope, and 900π square meters with a 30-meter rope.
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
you asked where I got the 5.
the 30m rope wraps around the corner at the other end of the 25m side, leaving a 5m section of rope to extend along the short side there.