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8. A box with no top is to be built by taking a 12 in-by-16 in 21. A city wants to build a new section of highway to link an ex-
sheet of cardboard, cutting x-inch squares out of each cor- isting bridge with an existing highway interchange, which
ner and folding up the sides. Find the value of x that maxi- lies 8 miles to the east and 10 miles to the south of the
mizes the volume of the box. bridge. The first 4 miles south of the bridge is marshland.
Assume that the highway costs AED 5 million per mile over
′′
′′
9. (a) A box with no top is built by taking a 6 -by-6 piece marsh and AED 2 million per mile over dry land. The high-
of cardboard, cutting x-in. squares out of each corner and way will be built in a straight line from the bridge to the
folding up the sides. The four x-in. squares are then taped edge of the marsh, then in a straight line to the existing
together to form a second box (with no top or bottom). Find interchange. (a) At what point should the highway emerge
the value of x that maximizes the sum of the volumes of the from the marsh in order to minimize the total cost of the
boxes. (b) Repeat the problem starting with a 4 in-by-6 in new highway? (b) How much is saved over building the
piece of cardboard. new highway in a straight line from the bridge to the in-
terchange?
10. Find the values of d such that when the boxes of exercise
9 are built from a din-by-4 in piece of cardboard, the maxi- 22. (a) After construction has begun on the highway in exer-
mum volume results from two boxes. cise 21, the cost per mile over marshland is reestimated
2
11. Find the point on the curve y = x closest to the point at AED 6 million. Find the point on the marsh/dry land
(0, 1). boundary that would minimize the total cost of the high-
way with the new cost function. If the construction is too
2
12. Find the point on the curve y = x closest to the point (3, 4). far along to change paths, how much extra cost is there
in using the path from exercise 21?
13. Find the point on the curve y = cos x closest to the point
(0, 0). (b) After construction has begun on the highway in exer-
cise 21, the cost per mile over dry land is reestimated at
14. Find the point on the curve y = cos x closest to the AED 3 million. Find the point on the marsh/dry land
point (1, 1). boundary that would minimize the total cost of the high-
way with the new cost function. If the construction is too
15. In exercises 11 and 12, find the slope of the line through the
given point and the closest point on the given curve. Show far along to change paths, how much extra cost is there
that in each case, this line is perpendicular to the tangent in using the path from exercise 21?
line to the curve at the given point.
16. Repeat exercise 15 for examples 7.3 and 7.4.
17. A soda can is to hold 12 fluid ounces. Suppose that the bot-
tom and top are twice as thick as the sides. Find the di- APPLICATIONS
mensions of the can that minimize the amount of material 23. Elvis the dog stands on a shoreline while a ball is thrown
used. (Hint: Instead of minimizing surface area, minimize x = 4 meters into the water and z = 8 meters downshore.
the cost, which is proportional to the product of the thick- If he runs 6.4 m/s and swims 0.9 m/s, find the place (y) at
ness and the area.) which he should enter the water to minimize the time to
18. Following example 7.5, we mentioned that real soda cans reach the ball. Show that you get the same y-value for any
have a radius of about 1.156 in. Show that this radius mini- z > 1.
mizes the cost if the top and bottom are 2.23 times as thick
as the sides.
Ball
19. A water line runs east-west. A town wants to connect two
new housing developments to the line by running lines x
from a single point on the existing line to the two develop-
ments. One development is 4 mi south of the existing line; Elvis y
Eliman
the other development is 5 mi south of the existing line and Eliman
5 miles east of the first development. Find the place on the z
existing line to make the connection to minimize the total
length of new line.
24. In the problem of exercise 23, show that for any x the opti-
20. A company needs to run an oil pipeline from an oil rig 25 mal entry point is at approximately y = 0.144x.
miles out to sea to a storage tank that is 5 mile inland. The
shoreline runs east-west and the tank is 8 mile east of the 25. Suppose that light travels from point A to point B as shown
in the figure. Assume that the velocity of light above the
rig. Assume it costs AED 50 thousand per mile to construct
Copyright © McGraw-Hill Education Copyright © McGraw-Hill Education construct the pipeline on land. The pipeline will be built in boundary is v . Find the total time T(x) to get from point
boundary line is v and the velocity of light below the
the pipeline under water and AED 20 thousand per mile to
1
2
′
A to point B. Write out the equation T (x) = 0, replace the
a straight line from the rig to a selected point on the shore-
square roots using the sines of the angles in the figure and
line, then in a straight line to the storage tank. What point
v
sin
on the shoreline should be selected to minimize the total
1
1
=
derive Snell’s Law
.
v
sin
cost of the pipeline?
2
2
297
297
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