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```MCV4U
date: _______________________________
MAX/MIN ON AN INTERVAL
MCV4U1
MORE OPTIMIZATION date: _________________________
1. Find the maximum and minimum value of each function on the given interval.
a) f x 
b) f x  ln x2  x 1,  2  x  2
ln x
, 1  x  e2
x
d) f x 
c) f x  x  2 e x , 0  x  2
3



8e x
,  ln 2  x  ln 4
e 2x  4

t 

20
2. The number of bacteria, N, in a culture at time t hours is N  200030  te . Find the



largest number of bacteria in the culture during the interval 0  t  50 .
3. The net monthly profit from the sale of a certain product is given (in dollars) by the formula

Px  106 1  x 1e 0.001x , where x is the number of items sold.
a) Find the number of items that yield the maximum profit. Assume that at most 2000 items can
be produced per month.
b) Repeat part (a) assuming that at most 500 items can be produced per month.
4. The concentration of two medicines in the bloodstream t hours after injection are c1t  tet
and c2t  t 2 e t . Which medicine has the larger maximum concentration?
5. The proportion of people who have responded to the advertisement of a new product after it

has been marketed for t days is found to be 0.71  e0.2t . The marketing area contains

10 million potential customers, and each response to the advertisement results in a profit to
the company of \$0.70 (on average), excluding the cost of advertising. The advertising costs
\$30 000 to produce and \$5000 per day to run.

0.2t
a) Find lim 0.71  e  and interpret the result.
t 
b)
c)
d)

e)
What percentage of potential customers have responded after 7 days of advertising?
Write the function Pt  that represents the net profit after t days of advertising.
What is the net profit after 28 days?
For how many days should the advertising campaign be run in order to maximize profits?
Assume an advertising budget of \$180 000. What is the maximum profit?
1
27
16
3
, min = 0 b) max = ln7, min = ln   c) max =
, min = 8 d) max = 2, min =
e
e
17
4
2. 74 715 at t = 20 hours
3.a) 1001 b) 500
4. the 2nd
5.a) 70% b) 52.7% c) Pt   10 7  0.7 2  1  e 0.2t  5000t  30000 d) \$4 711 880.47 e) 26 days,
\$4 712 968
1.a) max =


MCV4U1
OPTIMIZATION
1. Find two natural numbers whose sum is 16 and whose product is a maximum.
2. The sum of two natural numbers is 12. If the product of one number with the square of the
other is a maximum find the numbers.
3. Find two positive numbers whose sum is 15 if the sum of their squares is a minimum.
4. The perimeter of a rectangle is 24 cm. Find the dimensions of the rectangle of maximum area. What is
the maximum area?
5. The area of a rectangle is 64 cm 2 . Find the dimensions of the rectangle of minimum perimeter. What
is the minimum perimeter? No interval is required for this solution.
6. Three sides of a rectangular field are to be fenced with 420 m of fencing. Find the dimensions of the
field of largest area if the single fenced side must be at least as long as the two fenced sides.
7. A rectangular field is to be enclosed by a fence then subdivided into two areas by a fence parallel to
the shorter side. If 600 m of fencing material is available and each side must be at least 75 m in length,
find the largest possible total area that can be enclosed.
8. A rectangular field is to be enclosed by a fence then subdivided into three areas by fences parallel to
the shorter side. Find the dimensions of the largest total area that can be enclosed with 800 m of
fencing.
9. A rectangular box is made from a piece of cardboard which measures 48 cm by 18 cm by cutting equal
squares from each corner and turning up the sides. Find the maximum volume of such a box if:
a) the height of the box must be at most 3 cm.
b) the length and width of the base must be at least 10 cm.
10. A piece of paper for a poster has an area of 1 m 2 . The margins at the top and bottom are 8 cm
and at the sides are 6 cm. What are the dimensions of the sheet of paper which will maximize
the printed area of the page?
11. An open topped box has a square base and vertical sides. If the surface area of the box is 108
m 2 find the dimensions that will maximize the volume if the side of the base must be at least
4 m long.
12. A fence is built around a rectangular lot and the lot then subdivided into two lots by a fence
parallel to one of the sides. If the exterior fence costs \$2.50/m and the interior fence costs
\$1.00/m, find the dimensions of the lot of maximum area that can be fenced for a total cost of
\$480.
13. A closed cylindrical can is constructed from a fixed amount of material. Determine the ratio
of height to radius of the can with the maximum volume.
14. The volume of a closed cylindrical can is 25 m3 . Find the dimensions of the can with the
minimum surface area. No interval is required for this solution
15. The cost per cm2 of the top and bottom of a closed cylinder is three times the cost per cm2 of
the sides. Determine the ratio of the most economical dimensions of the container with
volume 12 cm 3 .
No interval is required for this solution.
16. The volume of a square based rectangular box is 252 dm3 . The construction cost of the
bottom is \$5.00 per dm2 , of the top is \$2.00 per dm2 and of the sides is \$3.00 per dm2 . Find
the dimensions that will minimize the cost if the side of the base must fall between 4 dm and
8 dm.
17. Find the dimensions of the rectangle of maximum perimeter that can be inscribed in a circle
18. Find the dimensions of the rectangle of maximum area that can be inscribed in an isosceles
triangle with base 40 cm and height 30 cm.
19. Find the dimensions of the cylinder of maximum volume that can be inscribed in a cone with
a diameter of 40 cm and a height of 30 cm.
20. Find the height and radius of the cylinder of greatest volume that can be inscribed in a sphere
21. A telephone company wants to run a cable from a point A on one bank of a river to a point B
on the opposite bank and 12 km down stream from point A. The river is 5 km wide. The
company can run the cable along the shore at a cost of \$1 000 per km and across the river at a
cost of \$2 000 per km. What lengths of cable should be run along the shore and under the
water to minimize the cost of the cable?
22. A power house cable runs from point P to a factory at point F located on the opposite bank of
a river 200 metres wide and located 400 metres down stream from P. If it costs \$12 per metre
to lay the cable under the water and \$6 per metre to run the cable along the bank find the
length of the cable which runs under water if the total cost is a minimum.
23. At 10:00A.M., a ship is located 40 nautical miles due west of a second ship. If the first ship
sails east at 20 knots and the second ship sails north at 10 knots when in the next two hours
will the two ships be closest together? Find the minimum distance between the two ships.
24. A sailing ship is located 25 km due south of a drifting vessel. If the ship sails north at 4 km/h
and the vessel drifts east at 3 km/h find the shortest distance between the vessels.
25. A wire of length 40 cm is cut into two pieces and bent to form a square and a circle. What
two lengths will minimize the total area of the two figures?
26. A rectangle is inscribed in the ellipse 9 x 2  16 y 2  3600 with its sides parallel to the
coordinate axes. Find the dimensions of the rectangle with the maximum perimeter.
27. A farmer estimates that if he digs his potato crop now he will have 120 bushels which he will
be able to sell for \$1 per bushel. If he waits, the crop will grow by 20 bushels per week while
the price will drop by \$0.10 per bushel per week. When should he dig his crop to maximize
his return?
28. A telephone company finds that there is a net profit of \$15 per phone if an exchange has
1000 subscribers. If there are more than 1000 their profits decrease by one cent for each
subscriber over that number. How many subscribers will maximize profit.
29. A school trip will cost each student \$15 if 150 students participate. However, the cost per
student will be reduced by \$0.05 for each student in excess of the 150. How many students
should make the trip to maximize total income?
30. Find the point on the graph of y  x which is closest to the point (1,0).
31. Find the minimum distance from the origin to the parabola y  5  x 2 .
1. 8,8
36 cm2
2. 8,4
3. 7.5,7.5
4. 6 cm by 6 cm,
5. 8 cm by 8 cm, 32 cm
6. 210 m by 105 m
7. 15 000 m 2
8. 200 m by 100 m
9a) 1 512 cm3
9b) 1 600 cm3
11. 6 m by 6 m by 3 m
12. 48 m by 40 m
10.
2 3
3
m by
m
3
2
3
25
100
m or
m h  3 100 m
2
2
13. h:r=2:1
14. r 
15. h:r=6:1
16. 6 dm by 6 dm by 7 dm
18. 15 cm by 20 cm
19. h  10 cm , r 
21.
3
40
cm
3
10 3
36  5 3
km across river and
km along shore.
3
3
23. 11:36AM , 8 5 n. mi.
24. 15 km
26. 32 units by 18 units
27. 2 weeks
29. 225 students
1 2
30.  ,

2 2 
17. 4 2 cm by 4 2 cm
20. h 
2 3R
,r 
3
400 3
m
3
160
40
cm ,
cm
25.
4
4
22.
28. 1 250 subscribers
31.
19
units
2
6R
3
```
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