Download 3.5 transformationsII

3.5 Shape-Changing Transformations II
Statistics
Questions are from pages 189 – 199 of the text. For some of the problems you will
have to refer to tables in the book.
D40.) For variable x taking on the integer values from 1 through 8, on the graph
below sketch the power function y = axb with a = 1 and b = 2. On the same
graph, sketch the exponential equation y = abx also with a = 1 and b = 2.
x
x2
2x
1 ______ ______
2 ______ ______
300
250
200
3 ______ ______
4 ______ ______
150
5 ______ ______
100
6 ______ ______
50
7 ______ ______
2
8 ______ ______
4
6
8
10
Discuss the differences between the exponential and power models.
P33.) Use the regression equation on page 189 to predict the weight of an alligator
that is 75 inches long.
P34.) Having a good measure of tidal velocity (the speed at which water depth
increases) in an estuary is critically important, especially during storms.
Tidal velocity is difficult to measure, but it is related to the depth of the
water. Thus, a good model of this relationship would allow scientists to
predict the velocity from measurements of water depth. Display 3.111 on
page 196 shows measurements of the depth of water (in meters) and tidal
velocities (in meters per second) for certain locations in an estuary.
a.) Describe the nature of the relationship between depth and velocity (is it
linear, exponential, or power).
b.) Fit an appropriate model that would allow the prediction of velocity
from depth.
First, have an equation with the logarithm of depth:
Next, have an equation with the depth:
P37.) For each of these relationships, first write the equation that relates x and y.
Then use the equation to find a power of y that you could plot against x in
order to get a linear plot.
a.) y is the area of a circle, and x is the radius of the circle.
b.) y is the volume of a block whose sides all have equal lengths, and x is
the side length.
c.) y is the volume of an 8-foot section of log with a circular cross section,
and x is the diameter of the log’s cross section.
E56.) More dying dice. Follow the same steps as P26 (from the previous
worksheet) for these numbers of surviving dice: 200, 72, 28, 9, 5, 2, and 1.
Use your data to estimate what the probability of “dying” was in order to
generate these numbers.
a.) Construct a scatterplot of the number of “live” dice versus the roll
number.
200
160
Population
120
80
40
0
1
2
3
Roll Number
4
5
6
b.) Transform the number of live dice using natural logs, and construct a
scatterplot of ln(dice) versus roll number.
Roll Number Population ln(population)
0
200
_______
1
72
_______
2
28
_______
3
9
_______
4
5
_______
5
2
_______
6
1
_______
5.0
ln (Population)
4.0
3.0
2.0
1.0
0
1
2
3
4
Roll Number
c.) What is the equation of the line by the least squares method?
d.) Transform this linear equation to the form y = abx
e.) Estimate the rate of “dying” for the dice.
5
6
Problem from an alternate text:
You have been put in charge of organizing a fishing tournament in which prizes
will be given for the heaviest fish caught. Since many of the fish will be caught
and released, you want to estimate the weight of the fish from its length. Since
length is one-dimensional and weight is three-dimensional, and since a fish 0 units
long would weigh 0 pounds, the weight of a fish should be proportional to the cube
of its length. To come up with a relationship, you have the following table of
length (cm) and weight (grams) for the fish type that the tournament is using.
Length Weight Length Weight Length Weight
(cm) (grams) (cm) (grams) (cm) (grams)
5.2
2
23.3
190
33.0
518
8.5
8
25.0
264
34.0
537
11.5
21
26.7
293
34.9
651
14.3
38
28.2
318
36.4
719
16.8
69
29.6
371
37.1
726
19.2
117
30.8
455
37.7
810
21.3
148
32.0
504
a.) You have decided to model using the relationship weight = a x length3
Take log10 of both sides and get in the form of a linear equation using
the properties of logarithms.
b.) Using the equation, a plot of (log length, log weight) does form a linear
relationship. By entering log length and log weight into the TI-83,
determine the equation for the Least Squares Regression Line.
c.) Using properties of logarithms, write this in the form:
weight = (constant) x lengthconstant power
d.) Using the result, predict the weight of a fish that has a length of 36 cm.