PRE-CALCULUS 11 Unit 1 – Day 7: MODELLING PROBLEMS

Quadratic functions have many applications and can be used to solve problems.

SOLVING PROBLEMS

First

You must understand the problem.

UNDERSTANDING THE PROBLEM

What is to be determined? What conditions must be satisfied to solve the problem? What information is given? Are there any restrictions?

Draw a diagram.

Second

Find the connection between the given information and what is to be determined.

You may be obliged to consider auxiliary problems if

an immediate connection cannot be found.

You should eventually obtain a plan to find the solution.

DEVISING A PLAN: DEFINE VARIABLES. WRITE AN EQUATION

Have you seen it before? Or have you seen the same problem in a slightly different form?

Do you know a related problem? Was it solved before? If so, could you use it? Could you use the result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

Look at the unknown! Do you know a theorem that could be useful? What quantities are needed? If the needed quantities are not given, can they be determined from the given information?

Could you restate the problem? Could you restate it still differently? Go back to definitions. Do you know a theorem that could be useful? Could you derive something useful from the data?

Third

Carry out the plan.

CARRY OUT THE PLAN: SOLVE THE EQUATION

Carrying out the plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct? Does your answer contradict any of the restrictions in the problem?

Fourth

Examine your solution.

LOOKING BACK: ANSWER THE QUESTION WITH UNITS

Can you check the result? Can you check the argument?

Can you derive the result differently? Can you see it at a glance?

Variables can be used to represent quantities in problems. If variables are used, you must define the variables on a diagram or with a written statement; it must be absolutely clear what each variable represents.

Unit 1 – Day 6: Modelling Problems Page 2 of 3

example: A bridge over a river is supported by a parabolic arch which is 40 metres wide at water level. The maximum height is 16 metres. Find the height of the arch above a point 10 metres from the centre of the river.

Unit 1 – Day 6: Modelling Problems Page 3 of 3

exercise: The cables of a suspension bridge hang in a curve which is roughly a parabola. The bottom of the cables reach the road level. If the supporting towers are 170 m apart and 40 m high, find the approximate height of the cable 15 m from where the cable touches the road level.