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 โ 19424.7
AC โ 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 ฯ/180. Then we have:
arc AC = 100 * (75ฯ/180) โ 130.9 feet
arc BC = 100 * (75ฯ/180) โ 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 โ 401.1 feet
Therefore, the distance around the sidewalk from path AC to path BC is approximately 401 feet.