Hi everyone, I have a WS due tomorrow, and would really appreciate some help.

You are taking a ride on a Ferris wheel that is 100 feet in diameter and has a bottom point that is 10 feet above the ground. Suppose that the wheel rotates four times every minute and from your friend's viewpoint on the ground, it is rotating in a clockwise direction.

1. Sketch your height y above the ground as it depends on the horizontal distance x from a vertical axis through center of the Ferris wheel.

2. Sketch your height y above the ground as a function of time t.

3. Find a formula for your height y as a function of t.

4. Sketch your horizontal distance x as a function of time t.

5. Find a formula for the horizontal distance x as a function of time t.

6. Find all intervals of time for which you are moving forward. Indicate these intervals on the graph in number 2.

7. Suppose your friends moves around the opposite side of the wheel so now it appears to him to be moving counter clockwise. From this new vantage point, how do your answers to number 6 change?

8. Find a formula relating your height y above the ground and horizontal distance x from the vertical axis through the center of the wheel.

I started by making a graph of the time (3.75, 7.5,11.25, and 15) for each rotation, the horiz distance from the y axis (don't know how to find) and the height (50,100,50,0). Any tips or solutions would be very much appreciated. Thanks

Sure, I'd be happy to help you step by step with your worksheet. Let's go through each question one by one.

1. Sketch your height y above the ground as it depends on the horizontal distance x from a vertical axis through the center of the Ferris wheel.
- To start, draw a vertical line representing the axis through the center of the Ferris wheel. On this line, mark a point representing the ground (0 feet) and another point representing the bottom point of the Ferris wheel (10 feet). Then, draw a horizontal line from the center to the right, representing the horizontal distance x. From the point on this line, mark points representing different heights y above the ground as the Ferris wheel rotates. Connect these points, forming a smooth curve that goes up and down.

2. Sketch your height y above the ground as a function of time t.
- On the y-axis of a new graph, mark points representing different heights y above the ground as the Ferris wheel rotates. On the x-axis, mark points representing time t. Connect these points, forming a smooth curve that goes up and down.

3. Find a formula for your height y as a function of t.
- The height y can be represented by a sine function. Since the Ferris wheel completes four rotations every minute, we know the period of the function is 1/4 minute. Additionally, the maximum height is 50 feet and the minimum height is -50 feet. Therefore, the formula for your height y as a function of time t is: y = 50sin(8πt) + 10.

4. Sketch your horizontal distance x as a function of time t.
- On a new graph, mark points representing different horizontal distances x from the vertical axis through the center of the Ferris wheel as the Ferris wheel rotates. On the x-axis, mark points representing time t. Connect these points, forming a smooth curve that goes clockwise.

5. Find a formula for the horizontal distance x as a function of time t.
- The horizontal distance x can also be represented by a sine function. Since the Ferris wheel completes four rotations every minute, we know the period of the function is 1/4 minute. Therefore, the formula for the horizontal distance x as a function of time t is: x = 50cos(8πt).

6. Find all intervals of time for which you are moving forward. Indicate these intervals on the graph in number 2.
- To determine the intervals of time for which you are moving forward, you need to find when the derivative of the position function (y or x) is positive. In this case, since the Ferris wheel is rotating clockwise, you are moving forward when the derivative of x (velocity) is positive. Calculate the derivative of x with respect to t: dx/dt = -400πsin(8πt). Set this derivative equal to 0 and solve for t to find the critical points. Then, analyze the sign of the derivative in the intervals between the critical points to determine when you are moving forward.

7. From your friend's new vantage point, where the Ferris wheel appears to move counterclockwise, your answers to number 6 will change. The intervals when you are moving forward will now be when the derivative of x (velocity) is negative. You can repeat the steps from question 6, but consider the negative sign.

8. The formula relating your height y above the ground and horizontal distance x from the vertical axis through the center of the wheel is: y = √(2500 - (x - 50)²) + 10.

I hope this helps you with your worksheet. Let me know if you have any further questions or need additional clarification!

Sure! Let's go through each question step by step.

1. Sketching your height y above the ground as it depends on the horizontal distance x from a vertical axis through the center of the Ferris wheel.
To do this, you can start by drawing a coordinate plane where the y-axis represents your height above the ground and the x-axis represents the horizontal distance from the vertical axis through the center of the Ferris wheel. Since the diameter of the Ferris wheel is 100 feet, the radius (distance from the center to the outer edge) would be 50 feet. Since the bottom point is 10 feet above the ground, you can mark that point on the y-axis. Then, using the radius as a guide, you can draw a semicircle from that point to represent one full rotation of the Ferris wheel. This will give you a rough sketch of your height above the ground as it depends on the horizontal distance x.

2. Sketching your height y above the ground as a function of time t.
Since the Ferris wheel rotates four times every minute, you can divide the time into quarters. For example, at t=0, t=0.25, t=0.5, and t=0.75 minutes, you will experience the same height above the ground. Therefore, your sketch will show your height y repeating every 0.25 minutes.

3. Finding a formula for your height y as a function of time t.
To find a formula for your height y as a function of time t, we need to take into account the rate at which the Ferris wheel rotates. Since it completes four rotations per minute, that means each rotation takes 1/4 of a minute. We can use the sine function to model this relationship. The formula for your height y as a function of time t is:
y(t) = 50 sin(8πt) + 10
This formula can be derived by using the fact that the radius of the Ferris wheel is 50 feet and the bottom point is 10 feet above the ground. The coefficient 8π comes from the fact that the Ferris wheel completes four rotations per minute, or eight π radians.

4. Sketching your horizontal distance x as a function of time t.
To sketch your horizontal distance x as a function of time t, you'll need to use the same time intervals as in question 2. Starting at t=0, you can measure the horizontal distance x from the vertical axis through the center of the Ferris wheel. Since the diameter of the Ferris wheel is 100 feet, the radius is 50 feet. As the Ferris wheel rotates, your horizontal distance x will change.

5. Finding a formula for the horizontal distance x as a function of time t.
To find a formula for your horizontal distance x as a function of time t, we can use the cosine function to model this relationship. The formula for x as a function of t is:
x(t) = 50 cos(8πt)

6. Finding all intervals of time for which you are moving forward and indicating these intervals on the graph.
To determine the intervals of time for which you are moving forward, you need to find the values of t where the derivative of x(t) is positive. Taking the derivative of x(t) with respect to t, we have:
x'(t) = -50 sin(8πt) * 8π
To find the intervals where x'(t) > 0, we can solve the inequality:
-50 sin(8πt) * 8π > 0
simplifying, we have:
sin(8πt) < 0
Since sin(t) < 0 in the interval (π, 2π), we can solve 8πt in that interval, i.e., π < 8πt < 2π. Simplifying it, we have:
1/8 < t < 1/4
Therefore, the intervals for which you are moving forward are (1/8, 1/4) minutes.

7. How do your answers to number 6 change from your friend's new vantage point?
When your friend moves around to the opposite side of the wheel, the direction of rotation appears to be counter-clockwise to him. Therefore, the intervals for which you are moving forward will have the opposite direction of rotation. In this case, the new intervals would be (5/8, 3/4) minutes.

8. Finding a formula relating your height y above the ground and horizontal distance x from the vertical axis through the center of the wheel.
Since the Ferris wheel is a circle, the relationship between your height y above the ground and horizontal distance x from the vertical axis through the center of the wheel can be described by the equation of a circle. The equation is:
x^2 + (y - 10)^2 = 2500
This equation comes from the Pythagorean theorem. Since the radius is 50 feet and the bottom point is 10 feet above the ground, the vertical distance from the center of the circle to any point (x, y) on the circle is (y - 10). The horizontal distance from the center to the point (x, y) is x. Using the Pythagorean theorem, we have x^2 + (y - 10)^2 = 50^2.

I hope this helps you in completing your WS! If you have any more specific questions, feel free to ask.