A pendulum of length 17.6 inches swing 1 degree and 5 minutes to each side of its vertical posisition. To the nearest hundreth of an inch, what is the length of the arc through which the end of the pendulum swings?
s = rθ
convert θ to radians and plug in the numbers
To solve this problem, we can use the formula for the length of an arc on a circle:
Length of arc = (angle in degrees / 360 degrees) * (circumference of the circle)
In this case, the angle is given as 1 degree and 5 minutes, which can be converted to decimal degrees as follows:
1 degree = 1 degree
5 minutes = 5/60 degrees = 0.0833 degrees
So the total angle is 1.0833 degrees.
The circumference of the circle is given by the formula:
Circumference = 2 * pi * radius
Since the pendulum length is given as 17.6 inches, half of this length is the radius:
Radius = 17.6 inches / 2 = 8.8 inches
Now, let's calculate the length of the arc:
Length of arc = (1.0833 degrees / 360 degrees) * (2 * pi * 8.8 inches)
Length of arc = (0.0030) * (2 * 3.14 * 8.8 inches)
Length of arc = 0.0030 * 54.9864 inches
Length of arc = 0.16496 inches
Rounding to the nearest hundredth of an inch, the length of the arc through which the end of the pendulum swings is approximately 0.16 inches.
To find the length of the arc through which the end of the pendulum swings, we can use the formula for the arc length of a circle sector:
Arc Length = (θ/360) * 2π * r
where θ is the angle in degrees, r is the radius (length of the pendulum), and 2π is approximately equal to 6.28.
In this case, the angle is given as 1 degree and 5 minutes. To convert the angle to decimal form, we need to convert the minutes to degrees. Since there are 60 minutes in a degree, 5 minutes is equal to (5/60) degrees, which is 0.0833 degrees.
Now we can calculate the arc length:
Arc Length = ((1 + 0.0833)/360) * 2π * 17.6 inches
Arc Length = (1.0833/360) * 6.28 * 17.6 inches
Arc Length ≈ 0.000192 * 6.28 * 17.6 inches
Arc Length ≈ 0.02199842 inches
Therefore, to the nearest hundredth of an inch, the length of the arc through which the end of the pendulum swings is approximately 0.02 inches.