Find the radius of a circle on which a central angle of 3 pie over 2 radians subtends an arc of 60 ft.
If a central angle theta = 3 pi / 2 then :
Circumference of a circle = 2 * r * pi
Ful circle have 360 ° = 2 pi radians.
Lenght of arc :
L = ( theta / 2 pi ) * 2 * r * pi
L = theta * r = 60
( 3 pi / 2 ) * r = 60 Multiply both sides by 2
3 pi * r = 120 Divide both sides by 3 pi
r = 120 / 3 pi = 40 / pi
To find the radius of a circle given a central angle and the length of its subtended arc, we can use the formula:
Arc length = radius × central angle
In this case, the central angle is 3π/2 radians and the arc length is 60 ft.
Substituting the values into the formula, we can solve for the radius:
60 ft = radius × (3π/2)
To isolate the radius, we divide both sides of the equation by (3π/2):
(radius × (3π/2)) / (3π/2) = 60 ft / (3π/2)
Simplifying on the left-hand side:
radius = 60 ft / (3π/2)
Next, we can simplify the expression on the right-hand side by multiplying the numerator and denominator by 2/π:
radius = (60 ft × 2) / (3π)
Further simplifying:
radius = 120 ft / (3π)
Finally, we can divide both the numerator and denominator by 3:
radius ≈ 40 ft / π
Therefore, the radius of the circle is approximately 40 ft/π.
To find the radius of a circle, we can use the formula:
radius = arc length / central angle
In this case, the given arc length is 60 ft, and the central angle is 3π/2 radians.
Let's plug these values into the formula:
radius = 60 ft / (3π/2)
We can simplify the formula by multiplying the numerator and denominator by 2/π:
radius = (60 ft * 2/π) / (3π/2 * 2/π)
Simplifying further:
radius = (120 ft/π) / (3)
radius = 40 ft/π
So, the radius of the circle is 40 ft/π.
First note that a whole circle measures 360 degrees or 2pi radians. The 3/2pi radians is just 3/4 of 2pi, Thus we can say that the length of arc given (the 60 ft) is just 3/4 of the circumference of the circle.
If 60 is just 3/4 of the whole, then 60/(3/4) = 80 ft is the circumference of the circle.
Recall that the circumference of circle is given by
C = 2*pi*r
where r = radius
Substituting,
80 = 2*pi*r
r = 80/(2*pi)
r = 40/pi = 12.74 ft
hope this helps~ :)