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example 7.5 to find any unrealistic assumptions we made. We study the problem of
designing a soda can further in the exercises.
In our final example, we consider a problem where most of the work must be done
numerically and graphically.

EXAMPLE 7.6 Minimizing the Cost of Highway Construction
The state wants to build a new stretch of highway to link an existing bridge with a
turnpike interchange, located 8 mi to the east and 8 mi to the south of the bridge.
There is a 5 mi wide stretch of marshland adjacent to the bridge that must be
crossed. (See Figure 4.90.) Given that the highway costs AED 10 million per mile to
build over the marsh and only AED 7 million per mile to build over dry land, how
far to the east of the bridge should the highway be when it crosses out of the marsh?

Bridge

5 Marsh

x 8 - x

3

Interchange
FIGURE 4.90
y A new highway
120 Solution You might guess that the highway should cut directly across the marsh,
so as to minimize the amount built over marshland, but this is not correct. We let x
represent the distance in question. (See Figure 4.90.) Then, the interchange lies
110
(8 − x) miles to the east of the point where the highway leaves the marsh. Thus, the
total cost (in millions of dirhams.) is
100
cost = 10(distance across marsh) + 7(distance across dry land).
x
2 4 6 8 Using the Pythagorean Theorem on the two right triangles seen in Figure 4.90, we
FIGURE 4.91 get the cost function
y = C(x) √ √
2
2
C(x) = 10 x + 25 + 7 (8 − x) + 9.
Observe from Figure 4.90 that we must have 0 ≤ x ≤ 8. So, we must minimize the
y
continuous function C(x) over the closed and bounded interval [0, 8]. From the
graph of y = C(x) shown in Figure 4.91, the minimum appears to be slightly less
10
than 100 and occurs around x = 4. We have
5 d [ √ ]
√
′
C (x) = 10 x + 25 + 7 (8 − x) + 9
2
2
x dx
7
2 4 6 8 = 5(x + 25) −1∕2 (2x) + [(8 − x) + 9] −1∕2 (2)(8 − x) (−1)
2
2
1
2
Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education -10 FIGURE 4.92 First, note that the only critical numbers are where C (x) = 0. (Why?) The only way
-5
7(8 − x)
10x
.
− √
= √
2
2
(8 − x) + 9
x + 25
′
′
′
to find these is to approximate them. From the graph of y = C (x) seen in
y = C (x)
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