David made a swing for his niece Sarah using ropes 2.4 m long, so that Sarah swings throuh an arc of length 1.2 m. Determine the angle through which Sarah swings, in both radians and degrees.
Have you learned the formula
arc length = r(theta) where theta is in radians ?
then theta = arclength/radius
= 1.2/2.4 radians
= 1/2 radians
we know
pi radians = 180º
1 radian = (180/pi)º
.5 radians = (90/pi)º = appr. 28.65º
28.6
To determine the angle through which Sarah swings, we can use the formula for the arc length of a circle, which is given by:
s = rθ
Where s is the arc length, r is the radius of the circle, and θ is the central angle in radians.
In this case, the arc length is given as 1.2 meters and the radius is half the length of the ropes, which is 2.4 / 2 = 1.2 meters.
Substituting these values into the formula, we have:
1.2 = 1.2θ
To solve for θ, divide both sides of the equation by 1.2:
θ = 1
So, Sarah swings through an angle of 1 radian.
To convert this to degrees, we can use the fact that π radians is equivalent to 180 degrees. Therefore, 1 radian is equal to:
1 radian * (180 degrees / π radians) ≈ 57.3 degrees
So, Sarah swings through an angle of approximately 57.3 degrees.
To determine the angle through which Sarah swings, we need to use the formula for the length of an arc:
Length of arc = radius * angle
In this case, the length of the arc is given as 1.2 meters, and the radius is half the length of the ropes, which is 2.4 meters.
So, let's substitute the values into the formula:
1.2 = 2.4 * angle
Now, let's solve for the angle:
angle = 1.2 / 2.4
angle = 0.5 radians
To convert radians to degrees, we can use the formula:
degrees = radians * (180 / pi)
Substituting the value we found for the angle in radians:
degrees = 0.5 * (180 / pi)
degrees ≈ 28.65°
Therefore, Sarah swings through an angle of approximately 0.5 radians or 28.65 degrees.