Lesson 3 The Bungee Jump Simulator Solidify Understanding

Jump Start

Write equations for both of the following graphs:

1.

2.

Learning Focus

Write equations that would produce given graphs created by adding or multiplying two functions together.

What does it mean to “model with mathematics,” and what can I attend to in order to make my models more precise?

How do I attend to the details in the features of a graph produced by combining two functions so that I can write the equation of the combined function?

Open Up the Math: Launch, Explore, Discuss

As a reward for helping the engineers at the local amusement park select a design for their next ride, you and your friends get to visit the amusement park for free with one of the engineers as a tour guide. This time you remember to bring your calculator along, in case the engineers start to speak in “math equations” again.

Sure enough, just as you are about to get in line for the Bungee Jump Simulator, your guide pulls out a graph and begins to explain the mathematics of the ride. To prevent injury, the ride has been designed so that a bungee jumper follows the path given in this graph. Jumpers are launched from the top of the tower at the left, and dismount in the center of the tower at the right after their up and down motion has stopped. The cable to which their bungee cord is attached moves the rider safely away from the left tower and allows for an easy exit at the right.

1.

Your tour guide won’t let you and your friends get in line for the ride until you have reproduced this graph on your calculator just as it appears in this diagram.

a.

Work with a partner to try and recreate this graph on your calculator screen. Make sure you pay attention to the height of the jumper at each oscillation, as given in the table.

horizontal	vertical	distance from midline
$0$	$120$	$40$
$1$	$48$	$32$
$2$	$105.6$	$25.6$
$3$	$59.52$	$20.48$
$4$	$96.38$	$16.38$
$5$	$66.89$	$13.11$
$6$	$90.49$	$10.49$
$7$	$71.61$	$8.39$
$8$	$86.71$	$6.71$
$9$	$74.63$	$5.37$
$10$	$84.30$	$4.30$
$11$	$76.56$	$3.44$
$12$	$82.75$	$2.75$
$13$	$77.80$	$2.20$
$14$	$81.76$	$1.76$
$15$	$78.59$	$1.41$
$16$	$81.13$	$1.13$
$⋮$	$⋮$

b.

Record your equation of this graph:

After a thrilling ride on the Bungee Jump Simulator, you are met by your host who has a new puzzle for you. “As you are aware,” says the engineer, “temperatures around here are very cold at night, but very warm during the day. When designing rides, we have to take into account how the metal frames and cables might heat up throughout the day. Our calculations are based on Newton’s Law of Heating.

“Newton found that while the temperature of a cold object increases when the air is warmer than the object, the rate of change of the temperature slows down as the temperature of the object gets closer to the temperature of its surroundings.”

Of course, the engineer has a graph of this situation, which he says “represents the decay of the difference between the temperature of the cables and the surrounding air.”

Your friends think this graph reminds them of the points at the bottom of each of the oscillations of the bungee jump graph.

2.

Using the clue given by the engineer, “This graph represents the decay of the difference between the temperature of the cables and the surrounding air,” try to recreate this graph on your calculator screen. (Hint: What types of graphs do you generally think of when you are trying to model a growth or decay situation? What transformations might make such a graph look like this one?)

Record your equation of this graph:

Ready for More?

Describe why Newton’s Law of Heating (or Newton’s Law of Cooling) includes a transformation of an exponential graph. How does the idea of an exponential function fit into this real-world context? What quantity is changing exponentially? How does this exponential change show up in the graph?

Takeaways

If I understand the behavior of different parent function types, I can combine functions to model complex real-world behavior.

For example,

I can cause the output of a function to increase or decrease exponentially by .
I can create periodic oscillations in a graph by .
I can identify constraints in a context that imply .
I can reflect a function horizontally by .
Since there are only a small number of parent function types, I can determine the functions that produce a function combination graph by .

Lesson Summary

In this lesson, we wrote equations to fit given graphs and tables of data that involved combining function types or reflecting functions to model the context.

Retrieval

1.

The graph tells the story of the fuel consumption (in miles per gallon, mpg) of my old car compared to the fuel consumption of my new car while traveling at various speeds ( $v mph$ ).

My old car averaged $15 mpg$ in the city and $25 mpg$ on the highway. The new car is predicted to average $32 mpg$ in city traffic and $42 mpg$ on the highway.

a.

If $f (v)$ is the function that describes the fuel consumption of my old car, what would be the function for the new car in terms of $f (v)$ ?

b.

Sketch the graph showing the fuel consumption of the new car.

c.

According to the graph, what is the most fuel-efficient speed to travel?

2.

Change each exponential expression to an equivalent expression involving a logarithm.

a.

$a^{3} = 5.7$

b.

$10^{6} = 1,000,000$

c.

$e^{5} = M$