A wheel with a radius of 3 cm is being pushed up a ramp at a rate of 7 cm per second. The ramp is 640 cm long, and 200 cm tall at the end. A point P is marked on the wheel. Point p is on top of a circle at the bottom of a right triangle aka ramp
a. Traveling at 7 cm/sec, how long does it take the wheel to complete one rotation?
b. What is the initial height of the center of the wheel?
c. Traveling at 7 cm/sec, in one second, how much does the height of the center of the wheel increase?
d. Write an equation for the height of the center of the wheel as a function of t, time in seconds.
e. Write an equation for the height of the point P above the ground as a function of t, time in seconds.
a. To find how long it takes the wheel to complete one rotation, we need to find the circumference of the wheel and divide it by the rate at which it is being pushed up the ramp.
The circumference of the wheel can be calculated using the formula C = 2πr, where r is the radius of the wheel. In this case, the radius is given as 3 cm. Therefore, the circumference is C = 2π(3) = 6π cm.
Since the wheel is being pushed up the ramp at a rate of 7 cm/second, it will take (6π)/(7) seconds for the wheel to complete one rotation.
b. The initial height of the center of the wheel can be determined by finding the height of the ramp at its starting position. The ramp is given to be 200 cm tall at the end, so at the starting position, the height will be 0 cm.
Therefore, the initial height of the center of the wheel is 0 cm.
c. In one second, the height of the center of the wheel will increase by the same rate at which it is being pushed up the ramp. In this case, the rate is given as 7 cm/second.
Therefore, the height of the center of the wheel will increase by 7 cm in one second.
d. The height of the center of the wheel as a function of time, t, can be represented by the equation:
h(t) = 0 + 7t
Here, h(t) represents the height of the center of the wheel at time t, and the equation shows that the height increases at a rate of 7 cm/second.
e. The height of point P above the ground can be determined by subtracting the height of the center of the wheel from the height of the ramp.
The height of the ramp at any point can be calculated using the equation of a right triangle:
h_ramp = (200/640) * x
Here, h_ramp represents the height of the ramp, x represents the distance traveled along the ramp, and 200/640 (or 5/16) is the slope of the ramp.
Since the ramp is a circular path, the distance traveled along the ramp, x, is related to the angle of rotation, θ, by the equation: x = rθ, where r is the radius of the wheel.
Therefore, the height of point P above the ground can be represented by the equation:
h_P(t) = (200/640) * (3θ)
However, we need to convert the angle of rotation into time, t. Since we know that it takes (6π)/(7) seconds for the wheel to complete one rotation (from part a), we can derive the equation:
h_P(t) = (200/640) * (3θ) = (200/640) * (3 * (2π/((6π)/(7))) * t)
Now simplifying:
h_P(t) = (200/640) * (3 * (2π/((6π)/(7))) * t) = (7/8) * t
Therefore, the height of point P above the ground as a function of time, t, can be represented by the equation:
h_P(t) = (7/8) * t