Related Rates Problem featuring a Ferris Wheel

Holydog23

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The text for the problem is in the picture, number 38. The accompanying figure can be found in another attached image. I need help getting started with figuring out the rate at which the passenger is rising. I have developed a few insights from the problem but these haven't exactly helped me get anywhere. For starters, I know that the radius of the wheel is 60 which implies that the circumference is C=2*pi*60=120*pi ft. We also know that a passenger starting at one point will go around to the exact same spot in two minutes. This means the passenger travels the length of the circumference every 2 minutes. or 60*pi ft every minute. Now the rate the first question inquires about is the rate at which the passenger is rising when she is 64 ft up in the air from the ground. I created a right angle triangle with the height as one of the sides (this can be imagined from the figure if you inspect it) and the radius of the wheel as its other side. The hypotenuse can be calculated but I see no use in that since the rate we are concerned with rates that have nothing to do with the hypotenuse from what I see.

Anyway, at this point I got stuck until I though about arc lengths. If the passenger starts at bottom of the ride (4 ft from the ground) and begins the trip, they are traveling an arc length that is a piece of the circumference (Im not sure if my reasoning is solid on this one but it's all I could do). We know arc length is s=r*theta where the angle is in radians. Since the radius of the wheel is constant we can substitute that in right way and we get s=60*theta. differentiating with respect to t results in the ds/dt = 60(dtheta/dt). at this point I made the assumption that the rate of change of the arc length is actually 60*pi ft/min which i derived above from dimensional analysis of the initial rate that was given in the text. I need clarification on this assumption.

Moving on, we get that the rate of change of the angle is simply pi radians/min. I am utterly lost after this point. I would really appreciate some advice on how to move forward or some corrections in my thinking. Some of my insights feel like I just made up my own math and fishy at best.
 
Maybe in trigonometry and precalculus you learned some things about the standard trig functions for sinusoidal events (angular velocity, phase shift, etc) and how to parameterize the motion of a point traveling around a circle as functions of time: x(t) and y(t)

x(t) gives the horizontal displacement of the point (rider).

y(t) + [height from ground to circle bottom] gives the vertical displacement of the point.

Once you've defined these parametric functions, their derivatives give you the horizontal and vertical velocity of the rider at time t.

That's where I would begin. :cool:
 
View attachment 10113View attachment 10114

The text for the problem is in the picture, number 38. The accompanying figure can be found in another attached image. I need help getting started with figuring out the rate at which the passenger is rising. I have developed a few insights from the problem but these haven't exactly helped me get anywhere. For starters, I know that the radius of the wheel is 60 which implies that the circumference is C=2*pi*60=120*pi ft. We also know that a passenger starting at one point will go around to the exact same spot in two minutes. This means the passenger travels the length of the circumference every 2 minutes. or 60*pi ft every minute. Now the rate the first question inquires about is the rate at which the passenger is rising when she is 64 ft up in the air from the ground. I created a right angle triangle with the height as one of the sides (this can be imagined from the figure if you inspect it) and the radius of the wheel as its other side. The hypotenuse can be calculated but I see no use in that since the rate we are concerned with rates that have nothing to do with the hypotenuse from what I see.

Anyway, at this point I got stuck until I though about arc lengths. If the passenger starts at bottom of the ride (4 ft from the ground) and begins the trip, they are traveling an arc length that is a piece of the circumference (Im not sure if my reasoning is solid on this one but it's all I could do). We know arc length is s=r*theta where the angle is in radians. Since the radius of the wheel is constant we can substitute that in right way and we get s=60*theta. differentiating with respect to t results in the ds/dt = 60(dtheta/dt). at this point I made the assumption that the rate of change of the arc length is actually 60*pi ft/min which i derived above from dimensional analysis of the initial rate that was given in the text. I need clarification on this assumption.

Moving on, we get that the rate of change of the angle is simply pi radians/min. I am utterly lost after this point. I would really appreciate some advice on how to move forward or some corrections in my thinking. Some of my insights feel like I just made up my own math and fishy at best.

you are right, rate of change of angle = pi radians/min. (2 revolutions in 2 min)
and also, rate of change in arc length = radius*rate of change in angle.
how fast the passenger is moving in x and y direction you get from where the passenger is on the wheel. You know the speed of the passenger around the wheel (their speed at that instant is tangent to the wheel). Then c2=x2+y2​.
 
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