I Was Going How Fast?
Accident investigators use the relationship s = 21d to determine the
approximate speed of a car, s mph, from a skid mark of length d feet, that it
leaves during an emergency stop. This formula assumes a dry road surface
and average tire wear.
1. A police officer investigating an accident finds a skid mark 115 feet long.
Approximately how fast was the car going when the driver applied the
brakes?
2. If a car is traveling at 60 mph and the driver applies the brakes in an
emergency situation, how much distance does your model say is required
for the car to come to a complete stop?
3. What is a realistic domain and range for this situation?
4. Does doubling the length of the skid double the speed the driver was
going? Justify your response using tables, symbols, and graphs.
Chapter 5: Square Root Functions
227
I Was Going How Fast?
Teacher Notes
Notes
Scaffolding Questions:
Materials:
•
Would the relationship between a skid mark and the
speed the car was going when it began to stop be a
functional relationship?
•
What factors influence realistic domain and range values
for the situation?
•
What is involved in stopping a car? Is it more than what a
skid mark shows?
•
Can a car stop without leaving a skid mark?
•
Besides speed, what else would contribute to the length
of a skid mark?
•
Did the function that was given relating skid mark to
speed take any other conditions into account?
•
What might you change about the equation to take some
of these other elements into account?
Graphing calculator
Algebra II TEKS Focus:
(2A.9) Quadratic and
square root functions. The
student formulates equations
and inequalities based on
square root functions, uses
a variety of methods to
solve them, and analyzes
the solutions in terms of the
situation.
The student is expected to:
(C) determine the
reasonable domain and
range values of square
root functions, as well as
interpret and determine
the reasonableness
of solutions to square
root equations and
inequalities.
(D) determine solutions
of square root equations
using graphs, tables, and
algebraic methods.
(F) analyze situations
modeled by square root
functions, formulate
equations or inequalities,
select a method, and
solve problems.
Additional Algebra II TEKS:
(2A.1) Foundations for
functions. The student uses
properties and attributes
of functions and applies
functions to problem
situations.
228
Sample Solutions:
1. The given value of 115 feet is the length of a skid mark,
d. Substitute for d in the formula.
s = 21d
s = 21* 115
s ≈ 49 mph
2. The given value of 60 mph is the approximate speed of
the car, s. Substitute for s in the formula and solve for d.
s = 21d
60 = 21d
3600 = 21d
d ≈ 171 feet
3. The function has a domain and a range of all positive
real numbers. The situation dictates that there is some
realistic maximum length and speed.
Chapter 5: Square Root Functions
I Was Going How Fast?
Teacher Notes
4. Doubling the length of the skid mark does not double the
speed. The table below shows that when the skid value is
doubled the speed value is not doubled. For example,
50 = 2(25) but 32.4 is not twice 22.9.
skid
speed
25 ft
50 ft
100 ft
200 ft
400 ft
22.9 mph 32.4 mph 45.8 mph 64.8 mph 91.7 mph
The graph also shows the function is not linear. Notice
the labeled points.
The student is expected to:
(A) identify the
mathematical domains
and ranges of functions
and determine reasonable
domain and range values
for continuous and discrete
situations.
(2A.9) Quadratic and square
root functions. The student
formulates equations and
inequalities based on square
root functions, uses a variety
of methods to solve them,
and analyzes the solutions in
terms of the situation.
To show this symbolically let s’ represent the speed when the
skid mark length is doubled.
s ' = 21 * (2d )
s ' = 2 21d
s ' = 1.414 21d
The speed is changed by a factor of approximately 1.414. It
is not doubled.
Extension Questions:
•
For what type of function would doubling the length of the
x-value double the y-value?
Linear functions of the form y = mx would be such that if
x is doubled then y will be doubled.
2y = m(2x)
Chapter 5: Square Root Functions
(B) relate representations
of square root functions,
such as algebraic, tabular,
graphical, and verbal
descriptions.
Connection
to TAKS:
Objective 2: The student
will demonstrate an
understanding of the
properties and attributes of
functions.
Objective 5: The student
will demonstrate an
understanding of quadratic
and other nonlinear functions.
Objective 10: The student
will demonstrate an
understanding of the
mathematical processes and
tools used in problem solving.
229
I Was Going How Fast?
Teacher Notes
•
Express the length of a skid mark as a function of speed.
s = 21d
s 2 = 21d
d =
s2
21
Note that the square root function and this quadratic function are inverses of each other
for positive values of s.
•
Investigators find that a car that caused an accident left a skid mark 143 feet long.
Damage to the car reveals that it was moving at a rate of 30 mph when it hit the other
car. How fast was the car going when it started to skid?
If the skidding car had not been stopped by hitting another car, it would have needed
302
another 21 ≈ 42.8 feet to skid before stopping. The total skid would have been (143
+ 43) feet. A skid mark this long would imply the car was going 21* 186 ≈ 62.5 mph
when it started to skid.
•
What, besides the actual braking distance, do you think affects the total distance that it
takes to stop a car in an emergency?
The distance that the car travels while the driver is reacting to the emergency situation
needs to be added to the braking distance to determine the total stopping distance.
•
There is a building on a corner of the highway that blocks a driver’s view for 150 feet. If
the speed limit on this stretch of highway is 55 mph, does a driver have enough time to
stop if there is a car broken down in the highway 150 feet around the corner?
Determine the distance to stop at 55 mph hour using the rule.
s = 21d
55 = 21d
552 = 21d
3025
≈ 144 feet
d =
21
Assuming the car could begin to stop the instant the driver saw the broken-down car, it
would take 144 feet to stop from a speed of 55 mph. However, other factors that are not
taken into account here, such as a driver’s reaction time, would increase this stopping
distance. Therefore, there is probably not enough room.
230
Chapter 5: Square Root Functions