Math 113 HW #2 Solutions
1. Exercise 1.2.18. The monthly cost of driving a car depneds on the number of miles driven.
Lynn found that in may it cost her $380 to drive 480 mi and in June it cost her $460 to drive
800 mi.
(a) Express the monthly cost C as a function of the distance driven d, assuming that a linear
relationship gives a suitable model.
Answer: Since we’re using a linear model, we want, first, to find the slope of the line.
Clearly, if the line goes through the points (480, 380) and (800, 460), then the slope of
the line is
460 − 380 80 1
= = .
800 − 480 320 4
Now we use the point-slope formula with the point (480, 380):
1 d
C − 380 = (d − 480) = − 120.
4 4
Hence,
d
C(d) = + 260.
4
(b) Use part (a) to predict the cost of driving 1500 miles per month.
Answer: Plugging in d = 1500 to the expression for C(d) yields
1500
C(1500) = + 260 = 375 + 260 = 635,
4
so we estimate that it will cost $635 to drive 1500 miles per month.
(c) Draw the graph of the linear function. What does the slope represent?
Answer: The graph is shown in Figure 1. The slope represents the marginal cost of
driving an additional mile.
500
400
300
200
100
0 100 200 300 400 500 600 700 800 900 1000
d
Figure 1: The cost function C(d) = 4 + 260.
1
, (d) What does the y-intercept represent?
Answer: The y-intercept, $260, gives the cost of owning a car which is independent of
the number of miles driven (for example, the cost of insurance would be included in this
cost). An economist might call this the “fixed cost of driving”.
(e) Why does a linear function give a suitable model in this situation?
Answer: A linear function gives a suitable model because we would expect the cost of
driving to be more or less proportional to the number of miles driven.
2. Exercise 1.3.14. Graph the function
y = 4 sin 3x
by hand, not by plotting points, but by starting with the graph of one of the standard
functions given in Section 1.2, and then applying the appropriate translations.
Answer: This graph will be like the graph of f (x) = sin x, except stretched vertically by a
factor of 4 and compressed horizontally by a factor of 3, yielding the graph shown in Figure 2.
4
2
-5 -2.5 0 2.5 5
-2
-4
Figure 2: The graph y = 4 sin 3x
3. Exercise 1.3.26. A variable star is one whose brightness alternately increases and decreases.
For the most visible variable star, Delta Cephei, the time between periods of maximum bright-
ness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness
varies by ±0.35 magnitude. Find a function that models the brightness of Delta Cephei as a
function of time.
Answer: Since the brightness alternately increases and decreases, we should model it by a
periodic function, which suggests either the sine or the cosine. Let’s try using sine. Then we
2
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