3.3 Optimization Problems
Chapter
Chapter 3
Section
3.3
Solutions 25 Videos

A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area.

2.20mins
Q1

Discuss the result of maximizing the area of a rectangle, given a fixed perimeter.

2.28mins
Q2

A farmer has 600 m of fence and wants to enclose a rectangular field beside a river. Determine the dimensions of the fenced field in which the maximum area is enclosed. (Fencing is required on only three sides: those that aren't next to the river.)

2.19mins
Q3

A rectangular piece of cardboard, 100 cm by 40 cm, is going to be used to make a rectangular box with an open top by cutting congruent squares from the corners.

Calculate the dimensions (to one decimal place) for a box with the largest volume.

5.15mins
Q4

A rectangle has a perimeter of 440 cm. What dimensions will maximize the area of the rectangle?

0.52mins
Q5

What are the dimensions of a rectangle with an area of 64 m^2 and the smallest possible perimeter?

3.03mins
Q6

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?

2.35mins
Q7

A net enclosure for practising golf shots is open at one end, as shown. Find the dimensions that will minimize the amount of netting needed and give a volume of 144 m^2. (Netting is required only on the sides, the top, and the far end.)

2.47mins
Q8

The volume of a square-based rectangular cardboard box needs to be 1000 cm^3. Determine the dimensions that require the minimum amount of material to manufacture all six faces. Assume that there will be no waste material. The machinery available cannot fabricate material smaller than 2 cm in length.

3.08mins
Q9

Determine the area of the largest rectangle that can be inscribed inside a semicircle with a radius of 10 units. Place the length of the rectangle along the diameter.

3.56mins
Q10

A cylindrical-shaped tin can must have a capacity of 1000 cm^3.

a) 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.

b) Express your answer for part a. as a ratio of height to diameter. Does this ratio meet the requirements outlined by the marketing department?

5.22mins
Q11

Determine the area of the largest rectangle that can be inscribed in a right triangle if the legs adjacent to the right angle are 5 cm and 12 cm long. The two sides of the rectangle lie along the legs.

3.20mins
Q12a

An isosceles trapezoidal drainage gutter is to be made so that the angles at A and B in the cross-section ABCD are each 120^{\circ}. If the 5 m long sheet of metal that has to be bent to form the open-topped gutter and the width of the sheet of metal is 60 cm, then determine the dimensions so that the cross-sectional area will be a maximum.

4.23mins
Q13a

Calculate the maximum volume of water that can be held by this gutter.

0.59mins
Q13b

The 6 segments of the window frame shown in the diagram are to be constructed from a piece of window framing material 6 m in length. A carpenter wants to build a frame for a rural gothic style window, where \triangle ABC is equilateral.

• Determine the dimensions that should be used to the six pieces so that the maximum amount of light will be admitted. Assume no waste material for corner cuts and so on.
4.18mins
Q14a

The 6 segments of the window frame shown in the diagram are to be constructed from a piece of window framing material 6 m in length. A carpenter wants to build a frame for a rural gothic style window, where \triangle ABC is equilateral.

• Would the carpenter get more light if the window was built in the shape of an equilateral triangle only? Explain.
1.08mins
Q14b

A train leaves the station at 10:00 am and travels due south at a speed of 60 km/h. Another train has been heading due west at 45 km/h and reaches the same station at 11:00am. At what time were the two trains closest together?

5.01mins
Q15

A north-south highway intersect an east-west highway at point P. A vehicle crosses P at 1:00 pm, travelling east at a constant speed of 60 km/h. At the same instant, another vehicle is 5 km north of P, travelling south at 80 km/h. Find the time when the two vehicles are closest to each other and the distance between them at this time.

5.09mins
Q16

You looked at two specific right triangles and observed that a rectangle with the maximum area that can be inscribed inside the triangle had dimensions equal to half the lengths of the sides adjacent to the rectangle. Prove that this is true for any right triangle.

1.38mins
Q17

Prove that any cylindrical can of volume k cubic units that is to be made using a minimum amount of material must have the height equal to the diameter.

4.07mins
Q18

A piece of wire, 100 cm long, is cut into two pieces. One piece is bent to form a square, and the other piece is bent to form a circle. Determine how the wire should be cut so that the total area enclosed is

a) a maximum

b) a minimum

6.21mins
Q19

Determine the minimal distance from point (-3, 3) to the curve given by y = (x -3)^2.

5.31mins
Q20

A chord joins any two points A and B on the parabola whose equation is y^2 = 4x. If C is the midpoint of AB, and CD is drawn parallel to the x-axis to meet the parabola at D, prove that the tangent at D is parallel to chord AB.

3.13mins
Q21

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 + 2y -10 = 0, find the rectangle with the maximum area.

2.25mins
Q22

The base of a rectangle lies along the x-axis, and the upper two vertices are on the curve defined by y = k^2 - x^2.Determine the dimensions of the rectangle with the maximum area.

3.11mins
Q23
Lectures 7 Videos

ex A farmer has 2400 m of fencing and wants to fence off a rectangular field that borders a straight river. He needs to fence along the river. What are the dimensions of the field that has the largest area?

5.42mins
2 Introduction to Optimization with example with boundaries

If 800 cm^2 of material is available to make a box with a square base, find the largest volume of the box.