Math 180 1.1 – 1.3: Watch Out For Falling Rocks
Source: Wendy Metzger’s “Problem Solving and Data Analysis Labs”
A national park visitor left a trail near the top of a waterfall to get a better look at the falls.
Against the complaints of her hiking companion, she started tossing rocks over the edge onto
the trail at the bottom of the falls, 92 feet below. Just then, a park ranger arrived at the scene.
The ranger cited the park visitor for climbing over a guard rail as well as endangering the lives
of people on the path below by throwing rocks. The park visitor argued that they ere small
rocks and she wasn’t throwing them hard enough to hurt anyone.
In this lab/groupwork you will approximate how fast one of the park visitor’s rocks was going
when it hit the ground near the bottom of the falls.
1. Draw a sketch to represent the problem situation.
2. The following table gives the height of the rock above the path at the base of the falls for
the first 0.80 seconds of its flight (data was collected using a Texas Instruments CBL). Time
is given in seconds after the rock was thrown, and height is given in feet above the ground.
Time
0.00
Height 96.56
0.10
97.62
0.20
98.39
0.30
98.85
0.40
99.00
0.50
98.85
0.60
98.37
0.70
97.62
0.80
96.57
a. Use your graphing calculator to make a scatter plot of the given data. Adjust your
graphing window so that you can see the shape of the data easily. You should see a
parabolic shape to the data. What is the vertex of the parabola?
b. Create a model for the data analytically (show your work in the space below) using the
vertex and one other data point from the table. (Hint: what is the standard form for a
quadratic function?).
3. According to your model, after how many seconds does the rock hit the ground?
4. Complete the following table using your model. Give the height to the nearest 0.01 feet.
Time
0.0
0.5
1.0
1.5
20.
2.5
2.9
Height
5. Graph the points in the table along with your model. Adjust your graphing window so that all
the points in the table show.
6. Notice that after 2.9 seconds, the rock is close enough to the ground by the bottom of the
waterfall to hit a person. Find the equation of the secant line that passes through the point
when the rock is thrown and 2.9 seconds after it is thrown. Show your work below. Graph
the secant line on the same screen as your model.
7. What is the average velocity (average rate of change of height over the change in time) of
the rock from the time it is thrown until 2.9 seconds after it is thrown?
8. Complete the following chart by finding the equation of the secant line to your model
through he endpoints of the given time interval and using the fact that the average velocity
during a time interval is the slope of the secant line for that interval.
Time Interval
1 to 2.9 sec
2 to 2.9 sec
2.5 to 2.9 sec
2.8 to 2.9 sec
2.89 to 2.9 sec
Equation of Secant Line
Avg. Velocity (feet/sec)
9. On the same graph with your model, graph each of the secant lines whose equations you
found in the chart above. Make a copy of this graph and turn it in with your lab/groupwork
report.
10. Use the results in numbers 8 and 9 above to estimate the speed at which the rock would
be traveling it hit a person on the ground. Give reasons for your answer. Give your estimate
in feet per second and in miles per hour.
11. If the rock struck a hiker on the path at the bottom of the waterfall, do you think the hiker
would be injured? Give reasons for your answer.
12. Write an explanation of average rate of change to a college algebra student who has not
done this lab. Describe what it means, how to compute it, and what it is used for, including
how you can use average rate of change to estimate rate of change at a single point on a
graph (i.e. instantaneous rate of change). You may illustrate your explanation with
examples or diagrams.