Learning Resources
Lesson
During this part of the chapter, you will be given the opportunity to use the mathematical concepts that you have learned. The Case Study problem will reflect the general nature of the chapter. This problem requires you to use the key skills, knowledge, principles, processes and concepts presented in this unit. You will be able to challenge these problems and may be requested to do part or all of the work. Consult your teacher.
The Extension section may require an extension of the material covered in the unit and will provide a strong challenge for you. These problems may involve applying procedures to situations that are similar to those in the chapter but unfamiliar or altering procedures to deal with new situations. You may need help to complete these.
- Case Study 1
- Case Study 2
- Extension 1
- Extension 2
Case Study 1
Before you begin you might like to review the solution to Question 23 on page 239 in your text. It is very similar to this case study.
The first step in the solution of the case study is to draw a diagram containing circles and triangles. The work in this unit can then be applied to the problem. Consider a side view of the cone with the ball inside. It would look something like the diagram below:
We need the dimensions of the cone, viz. its height (AD), slant height (AE) and the circumference of its base (GE is the diameter of the circular base).
- From the given information, how long is AF? AB? BC?
- BC ^ AE because the ball touches (is tangent to) the cone. Use DABC, trigonometry, and the lengths from step 1 to find mÐ BAC.
- From the given information, how long is AD (the altitude)?
- Use the DADE (note: AD ^ GE), trigonometry, and the information in steps 2 and 3 to find the length of AE, which is the slant height, and length of DE, which is the radius of the base of the cone.
- Calculate the circumference of the base of the cone using the radius found in step 4.
The second part of the problem deals with cutting a sector from a circle and using what is left to make the cone. Again a diagram should be drawn similar to the one below:
Think about cutting out the white sector and folding the yellow sector into a cone. It should be fairly obvious that the length of AE is both the radius of the circle and the slant height of the cone. It should also be obvious that 
is the circumference of the base of the cone. To see this more clearly, cut out a paper circle, remove a sector from it, and then fold what remains into a cone. Note the relation between the radius of the circle from which you cut the sector to form the cone and the slant height of the cone. Also note the relation between the length of the arc of the sector and the radius of the base of the cone.
- Using the value of the slant height found in step 4 above, calculate the circumference of the circle from which the sector is cut .
- Find the ratio of the circumference of the base of the cone (step 4) to the circumference of the circle (step 6).
- Use this ratio to calculate the size of the angle of the sector to be used to make the cone.
If you have the time and interest, use the above as a model to solve the Extension 1 problem.
Case Studies 2
Draw a sketch to represent the clock as below:
Placing this onto a co-ordinate plane will enable you to find the co-ordinates of each point by using 
where the central angles are multiples of 30°. Each point can be calculated manually or using the graphing calculator in parametric mode. Using calculator the parametric equations will be:
Set the table: TblStart = 0 with 
Tbl = 30 Copy and complete the table to find the co-ordinates:
| Time (o'clock) | Angle | x-co-ordinate | y-co-ordinate |
3 |
0° |
||
2 |
30° |
||
1 |
60° |
||
12 |
90° |
||
11 |
120° |
||
10 |
150° |
||
9 |
180° |
||
8 |
210° |
||
7 |
240° |
||
6 |
270° |
||
5 |
300° |
||
4 |
330° |
The chord between any two consecutive geraniums can be found by using the distance formula.
To determine the number of alyssum plants, the arc length can be found by taking 
th of the circumference. Once the number of these plants is determined you can find the central angle for each and create another table as above to find the co-ordinates for the alyssum plants.
Extension 1
This is a continuation of case study 1 - see the commentary for the case study and create a model that you can work with to find the required information.
Extension 2
This is an open ended question to complete case study 2; be creative in your design. Include all measurements in your design to make the placement of the plants very clear. Include the number of plants required for you design.
Activity
Complete case study and extension as directed by your teacher.
Test Yourself
As already indicated, your teacher may assign these case studies and/or extension activities as
- a replacement for the chapter project
- a homework assignment
- an assignment for evaluation purposes
- a portfolio item