A cow is tied on a 50-foot rope to the corner of a 20-ft by 50-ft rectangular building. The cow can graze on any of the grass it can reach. What is the area on which the cow can graze?
How did you get 120?
Draw a diagram of the building. The rope is attached at one corner. The cow can graze
a 3/4 circle of radius 50, where the 20-ft side of the building and the the closest 50-ft side of the building are two perpendicular radii.
When the cow reaches the far side of the building, 20 ft of rope lie along the short side of the barn, leaving her with only 30 ft of rope for the last 1/4 circle of grazing.
So, the total area she can graze is
3/4 * 2500π + 1/4 * 900π = 2100π ft^2
I used math but I don't feel like explaining it 😝
To determine the area on which the cow can graze, we need to determine the shape of the grazing area.
First, let's visualize the situation. We have a rectangular building with dimensions of 20 ft by 50 ft, and a cow tied to a corner of this building with a 50-foot rope.
The grazing area is the region within the reach of the rope while being tethered to the corner of the building. Since the cow cannot enter the building, the grazing area will be a combination of a circular area and a triangular area.
To calculate the grazing area, we need to calculate the areas of these two shapes separately and then add them together.
For the circular area:
1. The radius of the circular area is the length of the rope, which is 50 ft.
2. Using the formula for the area of a circle, A = πr², we can calculate the area of the circular portion within the reach of the rope.
For the triangular area:
1. This area is formed by one side of the rectangular building, the rope, and a line connecting the other end of the rope to the opposite corner of the building.
2. To calculate the height of the triangle, we can use the Pythagorean theorem.
- Suppose the longer side of the rectangular building is the base of the triangle, with a length of 50 ft.
- The length of the rope is the hypotenuse of the triangle, with a length of 50 ft.
- The height (remaining side of the triangle) can be calculated using the formula c² = a² + b².
- Using the Pythagorean theorem, we can solve for the height of the triangle.
3. Finally, using the formula for the area of a triangle, A = (1/2)bh, where b is the base of the triangle (50 ft) and h is the height calculated in the previous step, we can calculate the area of the triangular portion of the grazing area.
Once we have calculated the areas of the circular and triangular portions, we can add them together to obtain the total grazing area on which the cow can graze.