A cow is tethered by a piece of rope 50 meters long to the side of a rectangular barn. The rope is fastened to a hook (on the barn) that is located 20 meters from the corner along the longest side of the barn which measures 70 meters by 20 meters. Over how much ground (outside of the barn) can the cow graze? Please shiwnwork

To find the area over which the cow can graze, we need to determine the region enclosed by the 50-meter rope.

Let's visualize the situation:
1. We have a rectangular barn measuring 70 meters by 20 meters.
2. The rope is fastened to a hook located 20 meters from the corner along the longest side of the barn.
3. The cow is tethered to the hook by a rope that is 50 meters long.

To solve this problem, we can break it down into two parts:
1. Finding the circular region around the hook based on the length of the rope.
2. Finding the rectangular region enclosed by the rope and the barn.

1. Circular Region:
The rope forms a circle around the hook. The radius of this circle can be found using the Pythagorean theorem. Let's call this radius 'r':
r = √(50^2 - 20^2)
r = √(2500 - 400)
r = √2100

Now, we can calculate the area of the circular region using the formula A = πr^2:
A = π(√2100)^2
A ≈ 2100π square meters

2. Rectangular Region:
The rope also forms a right-angled triangle with one side being the shorter side of the barn (20 meters) and the other side being the distance from the hook to the corner of the barn (20 meters). Let's call the length of the third side 'd':
d = √(20^2 + 20^2)
d = √(400 + 400)
d = √800

Now, we can calculate the area of the rectangular region:
Area of the rectangular region = (70 - 20) * (√800)
Area of the rectangular region = 50 * (√800)
Area of the rectangular region ≈ 50 * 28.28
Area of the rectangular region = 1414 square meters

To find the total area over which the cow can graze, we add the area of the circular region and the area of the rectangular region:
Total area = Circular region area + Rectangular region area
Total area ≈ 2100π + 1414 square meters

So, the cow can graze over approximately 2100π + 1414 square meters of ground outside of the barn.

Draw a diagram. Clearly, the cow can graze 3/4 of a circle of radius 20m

So, the area is 3/4 * πr^2 = 300π m^2