Ok we are doing central angles and arcs and I can't figure out how to solve this problem: An arc is 70.7 m long and it subtends a central angle of 2ð/7. Find the diameter of the circle. How do you solve for this type of problem?
To solve this problem, you can use the formula relating the length of an arc to the measure of the central angle and the radius or diameter of a circle. The formula is:
Arc Length = (Central Angle / 360°) * Circumference
Given that the arc length is 70.7 m and the central angle is 2π/7, we can substitute these values into the formula to solve for the diameter of the circle.
Step 1: Find the circumference
We know that the circumference is equal to the arc length divided by the fraction of the circle it subtends. So, we have:
Circumference = (Arc Length) / (Central Angle / 2π)
Substituting the given values, we get:
Circumference = 70.7 m / (2π/7)
Circumference ≈ 70.7 m / (1.571)
Step 2: Find the diameter
The circumference of a circle is equal to π times the diameter, so we can write:
Circumference = π * Diameter
Substituting the value of the circumference we found in Step 1:
70.7 m / (1.571) = π * Diameter
Simplifying the equation, we get:
Diameter ≈ (70.7 m / (1.571π))
Step 3: Calculate the diameter
To find the diameter, we need to evaluate the expression:
Diameter ≈ 45.13 m
Therefore, the diameter of the circle is approximately 45.13 meters.