A circular arc has measure 22ft and is intercepted by a central angle of 27 degrees . Find the radius of the circle.
2πr/22 = 360/27
r = 22(360)/(54π) = 46.69 ft
check:
circum = 2π(46.69) = 293.3333
arc = (27/360)(293.3333 = 22
Well, let's call upon our trusty friend, the formula for the circumference of a circle: C = 2πr. We know that the measure of the circular arc is 22 ft, which means it represents a portion of the circumference. Since the central angle of the arc is 27 degrees, we can set up a proportion:
27 degrees / 360 degrees = 22 ft / 2πr
Simplifying the proportion, we get:
27 / 360 = 22 / (2πr)
Now, let's solve for r. Cross-multiplying, we have:
27 * 2πr = 360 * 22
54πr = 7,920
Dividing both sides by 54π, we find:
r = 7,920 / 54π
And voila! You have your answer. However, since you asked for the radius of the circle, I decided to calculate the answer for you. The exact value of the radius is approximately 23.157 ft. But hey, don't worry if you don't have π memorized to fifty digits. I don't either!
To find the radius of the circle, we can use the formula for the length of a circular arc:
Length of arc = radius * angle
In this case, we know the length of the arc is 22 ft and the angle is 27 degrees. Plugging these values into the formula:
22 ft = radius * 27 degrees
To find the radius, we need to isolate it on one side of the equation. Divide both sides of the equation by 27 degrees:
22 ft / 27 degrees = radius
Using a calculator, we can evaluate the left side of the equation:
0.814 ft/degree = radius
Therefore, the radius of the circle is approximately 0.814 ft/degree.
To find the radius of the circle, we can use the formula:
Length of the arc = (angle/360) * 2 * π * radius
Given that the length of the arc is 22ft and the central angle is 27 degrees, we can plug in the values into the formula and solve for the radius.
22ft = (27/360) * 2 * π * radius
First, simplify the fraction:
22ft = (0.075) * 2 * π * radius
Next, multiply by 2π:
22ft = 0.150π * radius
To solve for the radius, divide both sides of the equation by 0.150π:
radius = 22ft / (0.150π)
Now you can use a calculator to get the approximate value of the radius.