A dog is attached to a 2121-foot rope fastened to the outside corner of a fenced-in garden that measures 1818 feet by 2323 feet. Assuming that the dog cannot enter the garden, compute the exact area that the dog can wander.
Draw the figure. The dog can wander in 3/4 of a circle of radius 2122
and also, after it wraps past the short side of the fence, a 1/4 circle of radius 303.
So the total play area is 3/4 * π * 2121^2 + 1/4 * π * 303^2
To compute the exact area that the dog can wander, we need to determine the area of the region outside the garden but within the range of the dog's rope.
First, let's visualize the problem. The garden has dimensions of 18 feet by 23 feet, which means it is a rectangle. The dog is attached to a rope of 21 feet, fastened at the outside corner of the garden.
To understand the dog's wanderable area, we need to find the shape formed by the rope. Since the rope is fixed at the outside corner of the garden, it creates a quarter circle with a radius of 21 feet.
To find the area of the quarter circle, we need to use the formula for the area of a circle, but since it's only a quarter, we divide the result by 4.
The formula for the area of a circle is A = πr², where A represents the area and r represents the radius.
Substituting the values:
A = (π * (21 feet)²) / 4
Now, let's calculate it:
A = (π * 441 square feet) / 4
A = (3.14159 * 441 square feet) / 4
A ≈ 1089.348 square feet
So, the exact area that the dog can wander is approximately 1089.348 square feet.