A gazebo is located in the center of a large, circular lawn with a diameter of 200 feet. Straight paths extend from the gazebo to a sidewalk around the lawn. if two of the paths form a 75degree angle, how far would you have to travel around the sidewalk to get from one path to the other? Round to the nearest foot if necessary.
We can start by drawing a diagram of the scenario:
The two paths that form a 75-degree angle are labeled as AC and BC. We want to find the distance from A to B, which is the length of the path around the sidewalk.
We can use the fact that the lawn has a diameter of 200 feet to find the radius of the circle, which is 100 feet.
Next, we can use the law of cosines to find the length of AC, since we know the lengths of AB and BC. The law of cosines states that for a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab cos(C)
In our case, we can label AB as a, BC as b, and the angle between them (75 degrees) as C. Then we have:
AC^2 = AB^2 + BC^2 - 2AB*BC*cos(75)
AC^2 = 100^2 + 100^2 - 2*100*100*cos(75)
AC^2 ≈ 19424.7
AC ≈ 139.3 feet
Now we can use the fact that the paths extend radially from the center of the circle to the edge of the circle to find the length of the arc from A to C, and from B to C. We know that the radius of the circle is 100 feet, and the angle between each path and the radius at the center of the circle is 75 degrees. So the length of each arc is:
arc length = radius * angle in radians
We can convert the angle in degrees to radians by multiplying by π/180. Then we have:
arc AC = 100 * (75π/180) ≈ 130.9 feet
arc BC = 100 * (75π/180) ≈ 130.9 feet
Finally, we can add up the lengths of the two arcs and the length of AC to get the total distance around the sidewalk:
distance = arc AC + AC + arc BC
distance ≈ 401.1 feet
Therefore, the distance around the sidewalk from path AC to path BC is approximately 401 feet.
A gazebo is located in the center of a large, circular lawn with a diameter of 200 feet. Straight paths extend from the gazebo to a sidewalk around the lawn. if two of the paths form a 75degree angle, how far would you have to travel around the sidewalk to get from one path to the other? Round to the nearest foot if necessary.
183
262
131
3125
The correct answer is 262 feet.
To solve this problem, we can start by finding the length of the circular lawn's circumference. The circumference of a circle can be calculated using the formula:
Circumference = π * Diameter
Given that the diameter of the circular lawn is 200 feet, we can calculate the circumference as follows:
Circumference = π * 200
Circumference ≈ 628.32 feet (rounded to two decimal places)
Now, let's consider the two paths that form a 75-degree angle. Since the paths meet at the gazebo located in the center of the lawn, they divide the circumference into two parts.
To find the distance needed to travel around the sidewalk from one path to the other, we need to determine the fraction of the circumference covered by the angle formed by the two paths.
The angle of 75 degrees represents ((75/360) × 100)% of the full circle.
Angle fraction = (75/360) × 100
Angle fraction = 20.83% (rounded to two decimal places)
To find the length of the distance around the sidewalk between the two paths, we can multiply the circumference of the lawn by the angle fraction:
Distance around sidewalk = Circumference × Angle fraction
Distance around sidewalk ≈ 628.32 × 0.2083
Distance around sidewalk ≈ 130.81 feet (rounded to two decimal places)
Therefore, you would need to travel approximately 130.81 feet around the sidewalk to get from one path to the other.