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MCV4U
OPTIMIZATION PRESENTATION
PROBLEMS
Inscribing Problems
1.
Determine the area of the largest rectangle that can be inscribed in a right triangle with 5 cm and
12 cm long legs. The two sides of the rectangle lie along the legs of the triangle.
2.
A rectangle is inscribed in an isosceles right angles triangle of two sides each equals to 10 cm.
One side of the rectangle rests on the hypotenuse and the other two vertices on the two shorter
sides. Find the dimensions of the largest rectangle.
3.
A rectangle is inscribed in a circle x 2  y 2  9 . Find the dimensions of the largest rectangle and
its area.
4.
Determine the area of the largest rectangle that can be inscribed inside a semicircle with a radius
of 10 cm. Place the length of the rectangle along the diameter.
5.
A rectangle lies in the first quadrant, with one vertex at the origin and two of the sides along the
coordinate axes. If the fourth vertex lies on the line defined by x  2 y  10  0 , determine the
dimensions of the rectangle with the maximum area.
Revenue/Profit Problems
6.
An apple orchard now has 80 trees planted per hectare and the average yield is 400 apples per tree.
For each additional tree planted per hectare, the average yield per tree is reduced by
approximately four apples. How many trees per hectare will give the largest crop of apples?
7.
A fast food restaurant sells 80 sandwiches at $5 each daily. Survey shows that for each dollar
increase in price there will be 10 fewer sales, and for each dollar decrease in price there will be 10
more sales. If the cost of sandwich is $2 each, what is the optimal price that will produce the
maximum revenue?
8.
A real estate office manages 50 apartments in a downtown building. When the rent is $900 per
month, all the units are occupied. For every $25 increase in rent, one unit becomes vacant. On
average, all units require $75 in maintenance an repairs each month. How much rent should the
real estate office charge to maximize profits?
9.
The local soccer team plays at a stadium that seats 10 000 people. With ticket prices at $20, the
average attendance is 6000. For every $2 increase/decease in the ticket price, attendance falls/rises
by 300. At what price should tickets be sold to maximize the revenue?
10. A commuter train carries 2000 passengers daily from a suburb into a large city. The cost to ride
the train is $7.00 per person. Market research shows that 40 fewer people would ride the train for
each $0.10 increase in the fare, and 40 more people would ride the train for each $0.10 decrease.
If the capacity of the train is 2600 passengers, and at least 1600 passengers must be carried to run
the train, what fare should the railway charge to get the largest possible daily revenue?
Dimension Problems
11. What are the dimensions of a rectangle with an area of 64 m2 and the smallest possible perimeter?
12. A rancher has 1000 m of fencing to enclose two rectangular corrals. The corrals have the same
dimensions and one side in common. What dimensions will maximize the enclosed area?
13. A man wants to fence a rectangular field of area 3750 m2 and divide it in half with a fence parallel
to one of the sides of the field. What are the dimensions of the field that can minimize the cost of
the fence?
14. The total surface area of a square-based rectangular box is 4 ft2. Find the dimensions of the box
such that the volume is the maximum.
15. The total surface area of a closed cylindrical tank is to be 8 m2. What dimensions of the tank will
produce the maximum volume of the tank?
16. An open top rectangular box is to have a width of 5 cm and volume of 250 cm3. The cost is
$2/cm2 for the base material and $1/cm2 for the side material. What dimensions will have the
minimum cost of making the box?
17. A rectangular field of 100 m2 in area is to be enclosed by two kinds of fencing. The north-south
sides cost $10/m, and the east-west sides cost $5/m. What are the dimensions of the rectangular
field so that the cost is the minimum?
18. A rectangular field is to be fenced for a cost of $9000. Two opposite sides will be fenced using
standard fencing that costs $6/m, while the other two sides will require heavy-duty fencing that
costs $9/m. What are the dimensions of the rectangular field so that the area is the maximum?
19. Squares of equal length are cut from the four corners of a square sheet of side 12 cm. The four
sides are then folded to form an open top box. Find the length of the side of the squares cut that
gives a maximum volume.
20. The volume of an open top cylinder can is 355 cm3. Find the dimensions of the can that give the
minimum surface area.
21. A cylindrical-shaped tin can must have a capacity of 1000 cm3. Determine the dimensions that
require the minimum amount of tin for the can. (Assume no waste material.) According to the
marketing department, the smallest can that the market will accept has a diameter of 6 cm and a
height of 4 cm.
22. The total surface area of a square-based open top rectangular box is 12 square units. Find the
dimensions of the box such that the volume is the maximum.
23. A wire of length 80 cm is cut into two parts that are used to form the legs of a right triangle. How
should the wire be cut so that the area of the right triangle is the maximum?
24. A window of perimeter 8 m consists of a rectangle surmounted by an equilateral triangle. Find the
width of the window that admits most sunlight.