MATHEMATICS 201-103-RE
Differential Calculus
Martin Huard
Winter 2017
XIX – Optimization Problems
1. An all-news radio station has made a survey of the listening habits of local residents between
the hours of 5:00 P.M. and midnight. The survey indicates that the percentage of the local adult
population that is tuned to the station t hours after 5:00 P.M. is
P  t   18  2t 3  27t 2  108t  240
a) At what time between 5:00 P.M. and midnight are the most people listening to the station?
What percentage of the population is listening at this time?
b) At what time between 5:00 P.M. and midnight are the fewest people listening to the
station? What percentage of the population is listening at this time?
2. An inferior product with a large advertising budget sells well when it is introduced, but sales
fall as people discontinue use of the product. Suppose that the weekly sales S are given by
200t
S t  
2
t  2
where S is in millions of dollars and t is in weeks. After how many weeks will the sales be
maximized?
3. A farmer wishes to enclose a rectangular plot by a fence and the divide it into two plots by
another fence parallel to one of the sides. What are the dimensions of the largest area that can
be enclosed by using a total of 1800 meters of fencing?
4. Advertising fliers are to be made from rectangular sheets of paper that contain 400 square
centimeters of printed message. If the margins at the top and bottom are each 5 centimeters
and the margins at the sides are each 2 centimeters, what should be the dimensions of the fliers
if the total page area is to be a minimum?
5. A rectangular box with a square base is to be constructed so as to have a volume of 1 cubic
meter. Find the height and the length of the base of such a box if the amount of material for
used for its construction is to be minimal and the box has (a) an open top, (b) the box has a
closed top.
6. A cable television company has its master antenna located at point A on the bank of a straight
river 1 kilometer wide. It is going to run a cable from A to a point P on the opposite bank of
the river and then straight along the bank to a town T situated 3 kilometers downstream from
A. It costs $30 per meter to run the cable under the water and $18 per meter to run the cable
along the bank. What should be the distance from P to T in order to minimize the total cost of
the cable?
XIX – Optimization Problems
Math 103
7. A rectangle is to be inscribed in a right triangle having sides of length 6 meters, 8 meters and
10 meters. Find the dimensions of the rectangle with greatest area assuming the rectangle is
positioned so that it has a corner in the 90o angle.
8. Find the dimensions of the rectangle of greatest area that can be inscribed in a semicircle of
radius 100 centimeters.
9. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral
triangle of side 10 meters if one side of the rectangle lies on the base of the rectangle.
10. A closed cylindrical can is to have a surface area of 100 square centimeters. What should be
the height and radius of the cylinder in order for it to have the largest volume?
11. A rectangular storage container with an open top is to have a volume of 10 cubic meters. The
length of the base is twice the width. Material for the base costs $10 per square meter and the
sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
12. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter
of the window is 15 meters, find the dimension of the window so that the greatest possible
amount of light is admitted.
ANSWERS
105
1. a) 5:00 P.M 30%
b) 8:00 P.M
2. After 2 weeks
13.1%
8
3. 300m by 450m (where the fence dividing the two plots is 300 m long)
4. 4 1  10 cm by 10 1  10 cm

5.
a)


3

2 m  3 2 m  12 3 2 m
b) 1m  1m  1m
6. 94 km
7. 4m by 3m (where the 4m side is parallel to the 8m side in the triangle)
8. 100 2 m by 50 2 m
9. 5m by 5 2 3 m
10. radius:
5 6
3
12. Rectangle:
Winter 2017
cm
15
4 
m by
height:
30
4 
m
10 6
3
cm
11.
Circle of radius
15
4 
Martin Huard
3
36
2
m by
3
36 m by
5 3 36
18
m
m
2