TheFarmer Brown

Problem

ObjectivesWe are learning to:-

- use problems in different ways

- solve problems in many ways

- appreciate other solutions

- be more creative mathematicians

- think beyond one solution

- make connections within mathematics

The ProblemWhen Brownie, the chicken farmer

travelled to town at 30km/hr he noticed he arrived an hour too early. He noticed that when he travelled at 20km/hr he arrived an hour too late.

The Puzzles• How far was the return journey?• How fast should he travel to arrive on time?• How long did it take him to get to town?• How fast should he travel to arrive 2hrs late?• What excuse did he give to Mrs Brown?• What color was his tractor?

eggsactly!

Challenge

Solve one

of the

puzzles.

A Tablular Solutiont v=30 v=200 30 01 30 202 60 403 90 604 120 805 150 1006 180 1207 210 140

We are looking for the same distance two hours apart.Why?

Who would answer with this solution?

An Algebraic SolutionWe know speed = distance/time

So time = distance/speed

And that means… distance/30 plus 1 hr = distance/20 less 1 hr

WHY?

and solving this gives the ………...

A Graphical Solution

Time in hours

Distance in 20km

5

120

Why are there twodifferent startingpoints?

Slope = 20

Slope = 30

Another Graphical Solution

Time in hours

Distance in 20km

5

120The slopeof the red line approximatesthe speed.Prove this!

Slope = 30

Slope = 20

A Trigonometrical SolutionFrom the previous graph

slope m = Tan øWhere ø is the angle the line makes with the x-axis

So the slope of the red line is“the tangent of the mean of the angles of the slopes

of the two blue lines.”

Write that using Tan-1ø!

Another Algebraic Viewpoint

We know speed = dist/time

So speed x time = distance

Therefore

30(t-1) = 20(t+1) which is the distance d

solving this equation gives

t=5hrs and hence d = 120km

and the correct speed of 24km/hr.

A Calculus SolutionThe rate of change dy/dt = 30 km/hr and if

this is integrated with the boundary conditions that when t=0, y=0 and when t = t-1, y = d

We get the equation30(t-1)=d

Likewise 20(t+1)=d and we havethe same solution as the last slide.

Clever Algebra SolutionI know 30(t -1) = d ……(1)

and 20(t+1) = d ……(2)

If I triple eqn (2) and double eqn (1)

Then subtract I quickly get

d = 120km

Even more algebra!3x eqn (2) is 60t + 60 = 3d

And

2x eqn (1) is 60t - 60 = 2d

When we add these two equations

We get 120t = 5d

Or 120/5=d/t=v = 24km/hr

Another view!If we want to arrive on time

d/30 + 1 = d/20 - 1

Show this gives us the solution

d = 120km

Math SenseI travel at 20km/hr for longer than I travel at

30km/hr and yet I go the same distance.

So 3hr x 20km/hr and 2hr x 30km/hr all divided by the total time (3hrs + 2 hrs) I travel;

must be the average speed

v=24km/hr.

More BizzareIf I travelled there at 20km/hr and back at

30km/hr I would get home on time.30(t - 1) + 20(t + 1) = 2die 50t - 10 = 2d

25t - 5 = dor 5(5t - 1) = d

Which means d is a multiple of 5and the product of 5 and

one less than a multiple of 5.

Which means1 5 5x4 = 202 10 5x9 = 453 15 5x14 = 704 20 5x19 = 955 25 5 x 24 = 1206 30 5 x 29 = 145and so on…

are the ONLY possible solutions.120 is the first CM of 20 and 30.

Numbersense?The answer is between 20 and 30 and it

is closer to 20 than 30 because I spend longer travelling at 20.

It is closer in the ratio 2:3 or 4:6

The number I am looking for is 24.

Never only ONE wayThere are many different ways to solve

problems. We should seek other ways and share the results.

We must connect the table to the graph to the algebraic equation.

When we understand the properties we discover the beauty of mathematics.