A simple pulley with the given
radius r used to lift heavy objects is positioned 10 feet above
ground level. Given that the pully rotates �°, determine the height
to which the object is lifted.
(a) r � 4 in., � � 720°
(a)r=4in, θ=720°
720°(π/180°)
=4π
S=rθ
S=(4in)(4π)
S=16π
S≈50.27in
Well, what can I say? The object is definitely going up, that's for sure! But let's calculate how high it actually goes.
First, let's convert the radius to feet. Since 1 inch is equal to 1/12 feet, a radius of 4 inches is equal to 4/12 = 1/3 feet.
Now, let's find the height to which the object is lifted using the formula:
height = radius * angle in radians.
To convert the angle from degrees to radians, we need to multiply it by π/180. So, 720° * π/180 = 4π.
Plugging in the values, we get:
height = (1/3) * 4π = 4/3π feet.
Now, if you want an actual numerical answer, π is approximately 3.14. So, the height to which the object is lifted would be approximately:
height ≈ (4/3) * 3.14 feet ≈ 4.19 feet.
So, the object is lifted to a height of approximately 4.19 feet. Don't worry, no clown acrobatics needed here!
To determine the height to which the object is lifted, we can use the formula for the circumference of a circle, which is C = 2πr.
Given:
radius (r) = 4 inches
angle (θ) = 720 degrees
First, let's convert the angle from degrees to radians, because trigonometric functions typically use radians. To convert degrees to radians, use the formula: radians = degrees * π / 180.
θ = 720 degrees
θ = 720 * π / 180
θ = 12π radians
Now, we can calculate the circumference of the pulley using the formula:
C = 2πr
C = 2π * 4 inches
C = 8π inches
Since the pulley rotates 12π radians, we can determine the height to which the object is lifted using the formula:
height = C * θ / (2π)
height = 8π inches * 12π radians / (2π)
height = 96π² inches
We can convert inches to feet by dividing by 12:
height = 96π² inches / 12
height = 8π² feet
Therefore, the height to which the object is lifted is 8π² feet.
To determine the height to which the object is lifted, we need to calculate the circumference of the pulley and then multiply it by the number of rotations.
First, let's convert the radius from inches to feet:
r = 4 in. = 4/12 ft. = 1/3 ft.
Next, let's calculate the circumference of the pulley:
C = 2πr = 2π(1/3) = 2/3π ft.
Now, let's calculate the height to which the object is lifted:
Height = C * (θ/360°)
Height = (2/3π) * (720/360)
Height = (2/3π) * 2
Height = 4/3π ft.
Therefore, when the pulley rotates 720° with a radius of 4 in., the object is lifted to a height of approximately 4/3π ft.