9013 lesson 2.5 PROBLEM SOLVING

Example1

Three employees are hired to tar a rectangular parking lot of dimensions 20 m by 30 m. The first employee tars one piece and leaves the remaining shape, shown below, for the other 2 employees to tar equal shares. Show how they can share the job.

Answer

Draw a diagram
Guess and check
Use logic
Use formulas

A ₁ = (b)(h)

= (6)( 20)

=120cm

A ₂=bh/2

= (24)(20)

=240m

A2 has 120 m2 more the A1

∴take 60 m2 from A2 and add it to A1.

This can be rectangle of 15 × 4 taken from A2 (other configurations are possible).

The areas are now equal

A1 = 120 + 60 = 180 m2

A2 = 240 − 60 = 180 m

Example 2

Find the area of this patch of pavement.

Sample Solution

Answer

Draw a diagram
Make a scale model
Use a formula
Use technology (GSP®)
Use tools (geoboard)
A=A ₁ + A ₂A= (a+b)(h)

=bh+ bh                                                                                   2

           2

=3 (4)+3(4)                              OR                              =          (3+6)(4)

   2                                                                                               2

=6+ 12

=18 m²                                                                        =          18m²



With the given top and bottom lengths, you can use grid paper or a geoboard to determine a diagram that has a slanted side of 5 units, and thus determine the corresponding height. Once students know the height they can apply the area formula for a trapezoid; or separate the shape into a triangle and a rectangle and find each area and the sum of the areas.

Example

The diagram is a square inside a circle and a square outside the circle.

Which has the greater area?

– the space between the circle and the inside square (Diagram A)? Or – the space between the circle and the outside square (Diagram B)?
This question may be easier to do if you rotate the inside square 45°.

You can cut out the areas in question and see how they “fit” together. This will provide an acceptable answer and justification, particularly if done with several-sized circles.
Area DABC = 100.00 cm2

Area JMLK = 49.95 cm2

Area 􀀛NL = 78.22 cm2

(Area DABC) – (Area 􀀛NL) = 21.78 cm2

(Area 􀀛NL) – (Area JMLK) = 28.27 cm2

28.27 > 21.78

Therefore, the area between the circle and inside square is greater than the area between the circle and the outside square.

Note

NL is the GSP® symbol which refers to the area of the circle with radius NL.

Exercise
1. Use the Pythagorean relationship to determine that h = 4 m.
3. 2.The Little Can Company makes cylindrical cans with a height of 3 inches and radius of 1 inch. The entire lateral face is covered by a label. The paper for the labels is purchased in rolls 3 inches high. When unrolled the paper is 10 yards long. How many labels can be made from each roll, assuming the label does not overlap on the can?
  
  Show your work
5. 1. A field is 30m long and 15 m wide. Calculate
  2. a) the perimeter
  3. b) the area of the field.
  4 A farmer wants to fence off a piece of land as a paddock for his horses.
  
  There are, however, certain problems. One side of the land is cut off due to overhead power lines. On another side there is a very dangerous hole. See the diagram below to visualize the land. The shaded area represents the paddock that he can create.