5. A wheel 5.00 ft in diameter rolls up a 15.0° incline. How far above the base of the incline is the top of
the wheel after the wheel has completed one revolution?
A. 4.07 ft
B. 13.1 ft
C. 8.13 ft
D. 9.07 ft
one rotation is 5π ft
so the hypotenuse has length 5π
and
sin 15 = height/(5π)
height = 5π sin 15° = ......
a
To determine how far above the base of the incline the top of the wheel is after completing one revolution, we need to find the vertical distance the wheel has traveled.
First, let's find the circumference of the wheel using its diameter:
Circumference = π * diameter
Circumference = π * 5.00 ft
Circumference ≈ 15.71 ft
Since the wheel has rolled up a 15.0° incline, we can calculate the vertical distance using trigonometry. The vertical distance traveled is given by:
Vertical distance = Circumference * sin(angle)
Vertical distance = 15.71 ft * sin(15.0°)
Vertical distance ≈ 4.07 ft
Therefore, the top of the wheel is approximately 4.07 ft above the base of the incline after completing one revolution.
So, the correct answer is A. 4.07 ft.
To solve this problem, we need to break it down into steps:
Step 1: Determine the circumference of the wheel.
The circumference of a circle can be found using the formula C = 2πr, where r is the radius. In this case, the diameter of the wheel is given as 5.00 ft, so the radius would be half of that, which is 5.00 ft / 2 = 2.50 ft. Plugging this value into the formula, we find C = 2π(2.50) = 15.71 ft.
Step 2: Calculate the distance traveled by the wheel along the incline in one revolution.
To find this distance, we need to calculate the arc length of the circle. The arc length can be found using the formula S = θr, where θ is the angle in radians (15.0° needs to be converted to radians) and r is the radius. To convert degrees to radians, we use the formula θ (radians) = θ (degrees) × π/180. Plugging in the values, we find θ (radians) = 15.0° × π/180 ≈ 0.262 radians. Now we can calculate the distance traveled along the incline using S = 0.262 × 2.50 = 0.655 ft.
Step 3: Determine the height difference between the base and the top of the wheel after one revolution.
This height difference can be found using trigonometry. The height difference is the vertical distance covered by the wheel as it rolls along the incline. This distance can be found using the formula H = S × sin(θ), where S is the distance found in Step 2 and θ is the angle of the incline. Plugging in the values, we find H = 0.655 × sin(15.0°) ≈ 0.170 ft.
Therefore, the top of the wheel is approximately 0.170 ft above the base of the incline after one revolution.
None of the options provided match the calculated answer because there seems to be an error in the given problem.