John ties the leash of his dog to the corner of the doghouse. The dimensions of the dog house are 8ft x 8ft, and the leash is 10ft long. (hint: the chain is longer than the side of the dog house.)
a) About how many square feet can john's dog wonder?
b) If the dog walks in a circle, about how far can he walk?
make a sketch
Label the doghouse with corners A, B, C, and D
Can't the dog cover a semicircle with diameter 10 along AB and AD?
Can't he also come around B and cover a quarter-circle with diameter of 2 ?
Isn't the same thing possible at point D?
I am sure you can take it from there.
To answer these questions, we need to understand the geometry involved. Let's break it down step by step:
a) To find out how many square feet John's dog can wander, we need to determine the area within the radius of the leash. The leash is tied to the corner of the doghouse, so it creates a circular area. The radius of the circle is equal to the length of the leash, which is 10ft.
The area of a circle can be found using the formula: A = π * r^2, where A is the area and r is the radius. In this case, the radius is 10ft.
Plugging the values into the formula:
A = π * (10ft)^2
A ≈ 3.14 * 100ft^2
A ≈ 314ft^2
Therefore, John's dog can wander approximately 314 square feet.
b) To determine how far the dog can walk if it moves in a circle, we need to find the circumference of the circle. The circumference is the distance around the circle.
The circumference of a circle can be calculated using the formula: C = 2 * π * r, where C is the circumference and r is the radius. In this case, the radius is still 10ft.
Plugging the values into the formula:
C = 2 * π * 10ft
C ≈ 2 * 3.14 * 10ft
C ≈ 62.8ft
Therefore, John's dog can walk approximately 62.8 feet if it moves in a circle.