## Engage NY Eureka Math Algebra 2 Module 2 Lesson 2 Answer Key

### Eureka Math Algebra 2 Module 2 Lesson 2 Opening Exercise Answer Key

Suppose a Ferris wheel has a radius of 50 feet. We will measure the height of a passenger car that starts in the 3 o’clock position with respect to the horizontal line through the center of the wheel. That is, we consider the height of the passenger car at the outset of the problem (that is, after a 00 rotation) to be 0 feet.

a. Mark the diagram to show the position of a passenger car at 30-degree intervals as it rotates counterclockwise around the Ferris wheel.

Answer:

b. Sketch the graph of the height function of the passenger car for one turn of the wheel. Provide appropriate labels on the axes.

Answer:

c. Explain how you can identify the radius of the wheel from the graph in part (b).

Answer:

The graph of the height function for one complete turn shows a maximum height of 50 feet and a minimum height of – 50 feet, suggesting that the wheel’s diameter is 100 feet and thus its radius is 50 feet.

d. If the center of the wheel is 55 feet above the ground, how high is the passenger car above the ground when It is at the top of the wheel?

Answer:

The passenger car is 105 feet above the ground when it is at the top of the wheel. Since the graph displays the height above the center of the wheel, we would need to add 55 feet to 50 feet to get the height (in feet) above the ground.

### Eureka Math Algebra 2 Module 2 Lesson 2 Exercise Answer Key

Exercise 1.

Each point P_{1}, P_{2}, … P_{8} on the circle in the diagram to the right represents a passenger car on a Ferris wheel.

a. Draw segments that represent the co-height of each car. Which cars have a positive co-height? Which cars have a negative co-height?

Answer:

The cars corresponding to points P_{1}, P_{2}, P_{7}, and P_{8} have a positive co-height. The cars corresponding to points P_{3}, P_{4}, P_{5}, and P_{6 }have a negative co-height.

b. List the points in order of increasing co-height; that is, list the point with the smallest co-height first and the point with the largest co-height last.

Answer:

P_{5}, P_{4}, P_{3}, P_{6}, P_{2}, P_{7}, P1, P8

Exercise 2.

Suppose that the radius of a Ferris wheel is loo feet and the wheel rotates counterclockwise through one turn. Define a function that measures the co-height of a passenger car as a function of the degrees of rotation from the initial 3 o’clock position.

a. What is the domain of the co-height function?

Answer:

The domain of the co-height function is [0,3601], where we are measuring In terms of degrees of rotation.

b. What is the range of the co-height function?

Answer:

Because the radius is loo ft the range of the co-height function is [- 100,100].

c. How does changing the wheel’s radius affect the domain and range of the co-height function?

Answer:

Changing the radius does not change the domain of the co-height function.

The range of the co-height function depends on the radius; for a wheel of radius r, the range of the co-height function is [- r, r].

Exercise 3.

For a Ferris wheel of radius 100 feet going through one turn, how do the domain and range of the height function compare to the domain and range of the co-height function? Is this true for any Ferris wheel?

Answer:

The domain for each function is [0,3601], where rotations are measured in degrees. The range of each function is [- 100, 100]. For any Ferris wheel, the domain of the height and co-height functions is [0,360]. The range depends on the radius, r, of the wheel, but for both the height and co-height functions, the range is [- r, r]. Thus, the height and co-height functions for a Ferris wheel have the same domain and range.

### Eureka Math Algebra 2 Module 2 Lesson 2 Problem Set Answer Key

Question 1.

The Seattle Great Wheel, with an overall height of 175 feet, was the tallest Ferris wheel on the West Coast at the time of its construction in 2012. For this problem, assume that the diameter of the wheel is 175 feet.

a. Create a diagram that shows the position of a passenger car on the Great Wheel as it rotates counterclockwise at 45-degree intervals.

Answer:

The Great Wheel has a diameter of 175 feet, so the radius is 87.5 feet.

b. On the same set of axes, sketch graphs of the height and co-height functions for a passenger car starting at the 3 o’clock position on the Great Wheel and completing one turn.

Answer:

Below, the blue curve represents the height function, and the red curve represents the co-height function.

c. Discuss the similarities and differences between the graph of the height function and the graph of the co-height function.

Answer:

Both the height and co-height functions have the same domain, Lo, 3601, and range, [- 87.5,87.5]. Both functions have the same maximum value of 87.5 and minimum value of -87.5, but they occur at different amounts of rotation. When one function takes on a value of zero, the other either takes on its maximum value of 87.5 or its minimum value of – 87.5. The co-height function starts at its maximum value, and the height function starts at zero. The graph of the co-height function is the graph of the height function translated horizontally to the left by 90.

d. Explain how you can Identify the radius of the wheel from either graph.

Answer:

The radius of the wheel is the distance from the center of the wheel to a point on the wheel. We can easily measure this at one of the four points when the car is at the top or bottom of the wheel or at the far left or the far right. Thus, the radius is the difference between the maximum value of either function and zero, so the radius is the maximum value of either the height or the co-height function.

Question 2.

In 2014, the High Roller Ferris wheel opened in Las Vegas, dwarfing the Seattle Great Wheel with a diameter of 520 feet. Sketch graphs of the height and co-height functions for one complete turn of the High Roller.

Answer:

Question 3.

Consider a Ferris wheel with a 50-foot radius. We will track the height and co-height of passenger cars that begin at the 3 o’clock position. Sketch graphs of the height and co-height functions for the following scenarios.

a. A passenger car on the Ferris wheel completes one turn, traveling counterclockwise.

Answer:

b. A passenger car on the Ferris wheel completes two full turns, traveling counterclockwise.

Answer:

c. The Ferris wheel is stuck in reverse, and a passenger car on the Ferris wheel completes two full clockwise turns.

Answer:

Question 4.

Consider a Ferris wheel with radius of 40 feet that is rotating counterclockwise. At which amounts of rotation are the values of the height and co-height functions equal? Does this result hold for a Ferris wheel with a different radius?

Answer:

Consider the right triangle formed by the spoke of the wheel connecting the car to the center, the horizontal axis, and the perpendicular line dropped from the car’s position to the horizontal axis. If the value of the height and co-height functions are equal, then the legs of this triangle have the same length, meaning that it is an isosceles right triangle. There are four locations for such a triangle, with the passenger car being located in the first, second, third, or fourth quadrant.

However, in the second and fourth quadrants, either the co-height takes on a negative value or the height takes on a negative value, but not both. Thus, for the co-height and height to take on the same value, the passenger car must be in either the first or the third quadrant. In the first quadrant, the car has rotated through 45°, and in the third quadrant, the car has rotated through 180° + 45° = 225°.

The same result holds for a Ferris wheel of any radius.

Question 5.

Yuki is on a passenger car of a Ferris wheel at the 3 o’clock position. The wheel then rotates 135 degrees counterclockwise and gets stuck. Lee argues that she can compute the value of the co-height of Yuki’s car if she is given one of the following two pieces of information:

i. The value of the height function of Yuki’s car, or

ii. The diameter of the Ferris wheel itself.

Is Lee correct? Explain how you know.

Answer:

Lee is correct. Since Yuki’s car started at the 3 o’clock position and rotated 135°, then the ending position is in the second quadrant. The spoke of the Ferris wheel connecting her car to the center of the wheel makes a 45° angle with the horizontal, which creates a 45° – 45° – 90° triangle as shown in the diagram below. Then the height and the co-height at this position are equal, since the legs of an isosceles right triangle are congruent. Thus, if Lee knows the value of the height function of Yuki’s car, then she knows the value of the co-height at this position.

If Lee knows the diameter of the wheel, then she knows the radius, r, which is half of the diameter. Then she knows the length of a leg of an isosceles right triangle with hypotenuse of length r \(\frac{\sqrt{2}}{2}\) is r. Thus, if Lee knows the length of the diameter of the wheel, then she can calculate Yuki’s co-height.

### Eureka Math Algebra 2 Module 2 Lesson 2 Exit Ticket Answer Key

Zeke Memorial Park has two different-sized Ferris wheels, one with a radius of 75 feet and one with a radius of 30 feet. For either wheel, riders board at the 3 o’clock position. Indicate which graphs (a) – (d) represent the following functions for the larger and the smaller Ferris wheels. Explain your reasoning.

Answer:

Answer:

The maximum value of a passenger car’s height function over one turn will correspond to the highest point on the wheel, which means that the maximum value of the function is the radius of the wheel. Thus, the graphs that have a maximum value of 75 correspond to the larger Ferris wheel, and the graphs that have a maximum value of 30 correspond to the smaller wheel. Since the cars begin at the 3 o’clock position, the height graphs begin at height zero, while the co-height graphs begin with an initial co-height equal to the radius. Thus, graphs (a) and (d) correspond to the larger wheel, and graphs (b) and (c) correspond to the smaller wheel.

### Eureka Math Algebra 2 Module 2 Lesson 2 Exploratory Challenge Answer Key

Exploratory Challenge: The Paper Plate Model, Revisited

Use a paper plate mounted on a sheet of paper to model a Ferris wheel, where the lower edge of the paper represents the ground. Use a ruler and protractor to measure the height and co-height of a Ferris wheel car at various amounts of rotation, measured with respect to the horizontal and vertical lines through the center of the wheel. Suppose that your friends board the Ferris wheel near the end of the boarding period, and the ride begins when their car is in the three o’clock position as shown.

a. Mark horizontal and vertical lines through the center of the wheel on the card stock behind the plate as shown. We will measure the height and co-height as the displacement from the horizontal and vertical lines through the center of the plate.

Answer:

b. Using the physical model you created with your group, record your measurements in the table, and then graph each of the two sets of ordered pairs (rotation angle, height) and (rotation angle, co-height) on separate coordinate grids. Provide appropriate labels on the axes.

Answer:

While graphs may vary slightly from one group to the next, lead students to verbalize that it appears that the co-height graph is a horizontal translation of the height graph (and vice versa).

Encourage quantitative reasoning by asking students to relate features of the graph to the scenario of a car rotating around a Ferris wheel. The following questions can guide that discussion.

→ What do the zeros of the graph of the co-height function represent in this situation?

They represent the numbers of degrees of rotation where the passenger car is on the vertical line through the center of the wheel and has a horizontal distance from the center equal to 0. These are the highest and lowest positions of the car during the ride.

→ What does the vertical intercept of the graph of the co-height function represent in this situation?

It represents the radius of the wheel. At the outset of the ride, the car is at the 3 o’clock position, so it has rotated by 0 degrees, and the distance from the center is equal to the radius of the wheel.

→ How are the graphs of the height and co-height functions related to each other?

It looks like one graph is a horizontal translation of the other by 90°.

Closing

→ Why do you think we named the new function the co-height?

Answer:

Both functions measure a distance from the passenger car to one of the axes at various numbers of degrees of rotation of the wheel, so the horizontal measurements are closely related to the vertical measurements.

→ How are the graphs of these two functions alike? How are they different?

Answer:

The graph of the co-height function appears to be a horizontal translation of the graph of the height function. Assuming we create both functions from the same initial passenger car position on the same Ferris wheel, the two functions will have the same domain and the same range, but the values of the functions are not the same for the same amount of rotation. When one function has a value of zero, the other has either a maximum value of 1 or a minimum value of – 1.

→ What does a negative value of the height function tell us about the location of the passenger car at various positions around a Ferris wheel? What about a negative value of the co-height function?

Answer:

A negative value of the height function tells us the passenger car is below the center of the Ferris wheel.

A negative value of the co-height function tells us the passenger car is left of the center of the Ferris wheel.