Optimization Problems A
1.
A piece of wire 100 cm long is to be bent to form a rectangle. Determine the
dimensions of the rectangle of maximum area.
2. A farmer has 600 m of fence and he wants to enclose a rectangular field beside the
river on his property, by using the river as a natural fence. Find the dimensions of
the field so that a maximum area is enclosed.
3. Find the area of the largest rectangle that can be inscribed inside a semicircle with
radius of 10 units. Place the length of the rectangle along the diameter.
4. Find the lengths of the sides of an isosceles triangle which has a perimeter of 12 m
and a maximum area.
5. Two equal rectangular sand boxes having one side in common and a square floor each,
are made up of 96 m2 of material. What are the dimensions of each sand box if the
volume is to be a maximum?
6. Three hundred metres of fencing is available to enclose a rectangular field and
divide it into two smaller rectangular plots. What should the dimensions of the field
be for the area to be a maximum?
7. A manufacturer wishes to produce cylindrical fruit juice cans with a capacity of
250π mL. What dimensions will minimize the amount of material required for a can?
(1 mL = 1 cm3)
Answers:
1. 25 cm x 25 cm, Max area = 625cm2
2. 150 m x 300 m, Max area = 45 000m2
3. 𝑥 = 5 2, Max Area = 100 square units
4. All sides are 4 cm
5. 4 m x 4 m x
16
7
m, Max Volume =
512
7
2
= 73. 14𝑚
6. 75 m x 50 m, Max area = 3750 m2
7. r = 5 cm, h = 10 cm, Surface Area = 150π cm2
Optimization Problems - B
1.
The sum of two positive numbers is 20. Find two numbers such that,
a) The sum of their squares is a minimum.
b) The product of one and the square of another is minimum.
2.
Find the point on the parabola 𝑦 = 𝑥 that is closest to the point 𝐴(3, 0).
3.
Find a point on the line 3𝑥 + 4𝑦 − 25 = 0 which is closest to the origin.
4.
A rectangular poster paper has an area of 6912 𝑐𝑚 . Side margins are 8 cm each and top
and bottom margins are 6 cm each. Find the dimensions of the poster paper that gives the
maximum printing area.
5.
A man on an island is 4 km from the shore. He wants to go to a pub which is 8 km down the
shore. He can row at 3 km/h and walk at 5 km/h. Where should he land if he wants to reach
the pub as soon as possible?
6.
Find the area of the largest rectangle that can be inscribed inside a semi-circle with radius of
10 units. (Hint: Place the length of the rectangle along the diameter.)
2
2
Answers
1.
2.
3.
4.
5.
6.
a) 10 and 10
b)
(1, 1)
(3, 4)
96 cm by 72 cm
3 km away from the point on the shore closest to the boat
𝑥 = 5 2 and 𝐴 = 250 square units.
Optimization Problems - C
MCV4U1
1. Find two positive numbers whose product is 10 000 and whose sum is a minimum.
2. Find two positive numbers with product 200 such that the sum of one number and twice the
second number is as small as possible.
3. A rectangle has a perimeter of 100 m. What length and width should it have so that its area is a
maximum?
4. If 2700 cm2 of material is available to make a box with a square base and open top, find the
largest possible volume of the box.
5. A farmer wants to fence an area of 750 000 m2 in a rectangular field and divide it in half with a
fence parallel to one of the sides of the rectangle. How can this be done so as to minimize the
cost of the fence?
6. The Bouchard Soup Company estimates that the cost, in dollars of making x cans of pea soup is
C(x) = 48 000 + 0.28x + 0.000 01x2, and the revenue is R(x) = 0.68x – 0.00001x2
In order to maximize profits how many cans of pea soup should be sold?
7. A chain of stores has been selling a line of cameras for $50 each and has been averaging sales of
8000 cameras a month. They decide to increase the price, but their market research indicates
that for each $1 increase in price, sales will fall by 100. Find the price that will maximize their
revenue.
8. Find the points on the parabola y = 6 – x2 that are closest to the point (0, 3).
9. Find the point on the line y = 5x + 4 that is closest to the origin.
10. A boat leaves a dock at noon and heads west at a speed of 25km/h. Another boat heads north at
20 km/h and reaches the same dock at 1:00p.m. When were the boats the closest to each other?
Answers
1. 100, 100
2. 20, 10
3. 25 m x 25 m
4.
5.
6.
7.
8.
9.
10.
3
𝑥 = 30𝑐𝑚, 𝑉 = 13500 𝑐𝑚
500 2 𝑚 𝑏𝑦 750 2 𝑚
10 000
$65
(−−10 2. 5 2, 3. 5)𝑎𝑛𝑑 ( 2. 5 , 3. 5)
, 13
13
(
16
41
)
h after 12:00 noon (about 12:23 pm)
MCV 4U1
Optimization Problems – D
2
1. An open rectangular box (no top) with square base is to be made from 48 𝑚 of material.
What dimensions will result in a box with the largest possible volume? What is the
largest possible volume?
2. A container in the shape of a right circular cylinder with no top has a surface area of
2
3π 𝑚 . What height ℎ and what radius 𝑟 will maximize the volume of the cylinder?
What is the maximum volume?
3. Find the point (𝑥, 𝑦) on the curve 𝑦 = 𝑥 nearest to the point (4, 0). What is the
minimum distance? Include a sketch that clearly illustrates your solution.
4. Find the dimensions of the largest rectangle and the largest area that can be inscribed
inside a semi-circle with a radius of 8 m. Place the length of the rectangle along the
diameter. Illustrate with a sketch.
5. You are standing at the edge of a slow moving river which is 1 mile wide and wish to
return to your campground on the opposite side of the river. You can swim at 2 mph and
walk at 3 mph. You must first swim across the ricer to a point on the opposite bank.
From there you walk to the campground with is 1 mile from the point directly across the
river from where you start your swim. What route will take the least amount of time?
What is the minimum time?
Answers
1. 4m by 4 m by 2 m with a volume of 32 m3
2. Radius of 1m, height of 1m, volume of π.
3.
(
7
2
,
7
2
)
and distance of
15
2
4. Width = 4 2 and length = 8 2 with an area of 64 m2.
5.
2 5
5
miles down shore from the closest point across the river. It will take 0.71 hours.