Please help..
A wheel with radius 3 cm is being pushed up a ramp at a rate of 6 cm per second. The ramp is 730 cm long, and 150 cm tall at the end
A: Traveling at 6 cm/sec, how long does it take the wheel to complete one rotation?
B: What is the initial height of the center of wheel?
C:Traveling at 6 cm/sec, in one second, how much does the height of the center of 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
it goes 2 pi r cm in one revolution
2 pi (3) = 6 pi cm in a revolution
6 pi cm (1 second/6 cm) = pi seconds per rev
B. 3 cm
C. 6 cm *150/730
D. (6 cm *150/730)t
I have no idea where point P is but it is the center height from part D + a sin or cos function of time depending on where it is on the wheel.
I should be more specific about that sin or cos function arguement
sin or cos (2 pi t/T)
but we know that T is pi seconds
so
(2 t)
A: To find the time it takes for the wheel to complete one rotation, we need to find the circumference of the wheel. The circumference of a circle 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 of the wheel is:
C = 2π(3) = 6π cm
Since the wheel is traveling at a rate of 6 cm/sec, we can divide the circumference by the speed to find the time it takes to complete one rotation:
Time = Circumference / Speed = (6π) / 6 = π seconds
So, it takes the wheel π (approximately 3.14) seconds to complete one rotation.
B: The initial height of the center of the wheel can be determined by looking at the height of the ramp at the starting point, which is 0 cm. Therefore, the initial height of the center of the wheel is 0 cm.
C: The height of the center of the wheel increases as it travels up the ramp. Since the ramp is a straight incline with a height of 150 cm over a distance of 730 cm, we can use the ratio of the height to the distance traveled to find the increase in height.
Increase in height = (150/730) * 6 cm
So, the center of the wheel increases in height by approximately 1.23 cm in one second.
D: The equation for the height of the center of the wheel as a function of time can be expressed as:
Height (h) = (height increase per second) * time
In this case, the height increase per second is 1.23 cm, and time is represented by 't'. Therefore, the equation is:
h = 1.23t
E: To find the equation for the height of the point P above the ground as a function of time, we need to consider the height of the center of the wheel and the radius of the wheel. Since the center of the wheel is moving up the ramp, the height of point P above the ground is the sum of the height of the center of the wheel and the radius.
Height of point P = height of center of wheel + radius
Using the equation from part D, we can substitute the height of the center of the wheel as 1.23t and the radius as 3 cm:
Height of point P = 1.23t + 3 cm
So, the equation for the height of point P above the ground as a function of time is:
P(t) = 1.23t + 3 cm
A: To find the time it takes for the wheel to complete one rotation, we need to determine the circumference of the wheel. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Given that the radius of the wheel is 3 cm, we substitute this value into the formula.
C = 2π(3) = 6π cm
Now, we divide the circumference by the rate of 6 cm/sec to find the time it takes for one rotation.
Time = C/rate = (6π)/(6) = π seconds
Therefore, it takes π seconds for the wheel to complete one rotation.
B: To determine the initial height of the center of the wheel, we need to look at the dimensions of the ramp. The height of the ramp at the end is given as 150 cm. Since the wheel is being pushed up the ramp, the initial height of the center of the wheel will be 0 cm.
C: To find out how much the height of the center of the wheel increases in one second, we need to consider the slope of the ramp. The ramp has a height of 150 cm and a length of 730 cm.
The slope of the ramp is defined as the change in height divided by the change in distance traveled. So, in this case, the slope is 150 cm / 730 cm.
Now, we multiply the slope by the rate of 6 cm/sec to find the increase in height per second.
Increase in height/sec = slope * rate = (150/730) * 6 = 1.2339 cm/sec
Therefore, the height of the center of the wheel increases by approximately 1.2339 cm per second.
D: The equation for the height of the center of the wheel as a function of time can be derived using the slope of the ramp. The slope is given by the equation:
slope = change in height / change in distance
Since the height increases uniformly with time, we can express the change in height as a function of time. Let's denote the height of the center of the wheel as h(t), where t is the time in seconds.
The change in height is equal to the rate of increase (1.2339 cm/sec) multiplied by the time:
change in height = (1.2339 cm/sec) * t
The change in distance is equal to the rate of 6 cm/sec multiplied by the time:
change in distance = (6 cm/sec) * t
Now, substituting these values into the equation for slope:
slope = (1.2339 cm/sec) * t / (6 cm/sec) * t
Simplifying the equation, we find:
slope = 1.2339 / 6 = 0.20565
Therefore, the equation for the height of the center of the wheel as a function of time is:
h(t) = (0.20565 cm/sec) * t
E: To write an equation for the height of the point P above the ground as a function of time, we need to consider the radius of the wheel. We know that the center of the wheel has a height of h(t) as given in the previous equation.
The height of point P above the ground is equal to the height of the center of the wheel plus the radius of the wheel.
Therefore, the equation for the height of point P as a function of time is:
P(t) = h(t) + 3 cm
Substituting the value of h(t) from the previous equation, we have:
P(t) = (0.20565 cm/sec) * t + 3 cm