MthSc 207
Chapter 9 Review
D(g,p)
1. D(g,p) represents the demand (in thousand passengers) for air travel between the US and
Europe when g is the gross national product (in billion dollars) and p is the price of a
roundtrip ticket (in dollars).
a) Find a cross-sectional model that could be used to find D(2900, 725). Define the model
completely.
b) Using your model from part a, find D(g,p) when g = 2900 and p = 725.
c) Use the derivative of the model that you found in part a to find
∂D
∂p
. Give units
(2900,725)
with your answer.
d) Draw the D(g,p) = 6000 contour on the table at the top of the page.
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1
e) The multivariable equation for this function is D(g,p) = 15.4406g1.905p-1.247 . (This will be
different from the cross-sectional model that you found in part a.) Use this model to
find the formulas for Dg and Dp.
f) Complete the chart below using the equation from part e. Then draw the 6000 contour
curve on the axes provided. Label the axes.
D(g,p) = 6000
g
p
1400
1700
2000
2300
2600
2900
2000
1500
1000
925
500
0
1400
1700
2000
2300
2600
2900
g) Draw the tangent to the 6000 contour curve at g = 2000, p = 925.
h) Find the slope of the tangent line that you drew in part f. Use math8 on your calculator
or use the partial derivative formulas that you found in part e.
i) If the gnp falls from 2000 to 1800, estimate the change in the price necessary to
compensate and keep the demand at 6000. Do not use solve. Estimate using your answer
to part h. Show your work.
j) How does the 6000 contour curve that you graphed in part e compare to the 6000
contour curve that you drew on the table in part c?
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2
2. Your body-mass index is a measurement of how thin you are compared to your height. A
person’s body-mass index1 is given by
B(h, w) =
0.45w
points or B(h, w) ≈ 697.5wh–2 points
0.00064516h2
where h is your height in inches and w is your weight in pounds.
Table: Body-mass index of a person h inches tall who weighs w pounds
height h in inches
60
62
64
66
68
70
72
74
76
78
90
17.44
16.33
15.33
14.41
13.58
12.81
12.11
11.46
10.87
10.32
100
19.38
18.15
17.03
16.01
15.08
14.23
13.45
12.74
12.08
11.46
weight
110
21.31
19.96
18.73
17.61
16.59
15.66
14.80
14.01
13.28
12.61
w in
120
23.25 21.77 20.43
19.21
18.10
17.08
16.15
15.28
14.49
13.76
pounds
130
25.19 23.59 22.14 20.82
19.61
18.51
17.49
16.56
15.70
14.90
140
27.13 25.40 23.84 22.42
21.12
19.93
18.84
17.83
16.91
16.05
150
29.06 27.22 25.54 24.02 22.63 21.35
20.18
19.11
18.11
17.20
160
31.00 29.03 27.25 25.62 24.13 22.78 21.53 20.38 19.32
18.34
170
32.94 30.85 28.95 27.22 25.64 24.20 22.87 21.65 20.53 19.49
180
34.88 32.66 30.65 28.82 27.15 25.62 24.22 22.93 21.74 20.64
190
36.81 34.48 32.35 30.42 28.66 27.05 25.56 24.20 22.94 21.78
200
38.75 36.29 34.06 32.02 30.17 28.47 26.91 25.47 24.15 22.93
210
40.69 38.10 35.76 33.63 31.68 29.89 28.26 26.75 25.36 24.08
220
42.63 39.92 37.46 35.23 33.19
230
44.56 41.73
240
46.50 43.55 40.87 38.43 36.20 34.16 32.29 30.57 28.98 27.51
250
48.44 45.36 42.57 40.03 37.71 35.59 33.64 31.84
31.32 29.60 28.02 26.57 25.22
39.17 36.83 34.69 32.74 30.95 29.30 27.77 26.37
30.19 28.66
Check your calculator typing of the function with B(54, 203)= 48.55719588.] Units of
measure should be included with all numerical answers, models, and derivative formulas.
Round all answers to three decimal places unless otherwise directed.
Revised: August 2003
3
1. Find the body-mass index for Sparky who is 5’10” tall and who weighs 200 pounds. Round
the BMI to the nearest whole number.
Answer = _____________
B(h, w)
2.points
Locate
Sparky’s height, weight, and
45 BMI on the contour graph to the right
35 by placing a visible dot on the contour
25 graph at the proper location.
15
60
64
68
72
h
inches
76
100
140
250
220
180
w
pounds
w
pounds
250
45
40
35
220
30
25
180
20
140
100
15
60
64
68
72
76
h
inches
3. Draw, on the table on the previous page, the contour curve where B(h, w) equals the value
in question 1. (Use a pencil, and be sure to “smooth out” the contour.)
4. Use the equation for B(h, w) and the solver on your calculator to draw the contour curve
where B(h, w) equals Sparky’s BMI value in question 1. Show a table of at least 6 values in
your work. Report your values rounded to the nearest tenth.
w
B(h,w) = _____
h
w
(pounds)
62
64
66
68
70
72
h
(inches)
5. Suppose Sparky grows 2 inches in height. Locate Sparky’s current height on the contour
curve that you drew in #4. Show the 2” growth as an arrow beginning on the contour curve.
Show graphically, with another arrow drawn from the tip of the first arrow, the change in
weight required to maintain Sparky’s current body-mass index. Estimate from the graph
the change in the weight.
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6. Find the formulas for the partial derivatives of B(h, w) with respect to the inputs h and w.
Bh
Bw
7. Evaluate the partial derivatives of B(h, w) at Sparky’s original height and weight
(5’ 10”, 200). Include units. If you use your calculator, show what you’ve typed on your
calculator.
Bh
Bw
8. Find
dw
at the point on the contour curve corresponding to Sparky’s original height and
dh
weight. Include units.
9. Draw the tangent line at Sparky’s original height and weight on the contour curve you
graphed in question 2.
Is the slope of the tangent line positive or negative?
Does the slope of the line appear to be the value you found for
10. Use
dw
?
dh
dw
to estimate the weight change needed to compensate for a 2-inch growth if
dh
Sparky’s body-mass index is to remain constant. How does this value compare with the
estimated value you found in question 3?
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