Fencing Prblem

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Introduction

A Farmer has a plot of land. She has only 1000m of fencing. She wants to fence off a plot of land. She is not concerned about the shape but the perimeter has to be exactly 1000m. The fenced off area has to be the maximum contained area. I am going to investigate the shape or shapes that could be used to fence in the maximum area using exactly 1000 metres of fencing each time.

Firstly, I am going to investigate 3 - sided shapes - triangles, 4 – sided shapes-quadrilaterals and 5 sided shapes-pentagons. From there on I am going to gather the results and see if I need to investigate any further.

TRIANGLES

In first part of my coursework I am going to investigate different types of triangles.

Firstly I will look at Isosceles triangles.

The formula for triangles:

There are two ways of working out the area. You will get the same answer with either formula.

  • Area = ½ x base x height
  • Area = base x height

2

Triangle 1

                             450m                        450m

                                        

    100m

I divide the Isosceles triangle

450 m2 - 50m2 = Height m2

202,500 – 2,500 = 200,000

Height = square root of 200,000

Height = 447.21 to two decimal places.

To calculate the area of this Isosceles triangle we calculate:

Area = ½ x base x height

Area = ½ x 100 x 447.21

 

Area = 22,361 m2 (meter squared) to the nearest whole number.

Triangle 2

I chose a different size isosceles triangle to continue my investigation

 

                             425m                        425m

                                        

   150m

I divide the Isosceles triangle

425 m2 - 75 m2 = Height m2

140625 – 5625 = 175000

Height = square root of 175000

Height = 418.33 to two decimal places

To calculate the area of this Isosceles triangle we calculate:

Area = ½ x base x height

Area = ½ x 150 x 418.33

Area = 31,375 m2 (meter squared) to the nearest whole number.

Triangle 3

I chose another size isosceles triangle to continue my investigation.

                             400m                        400m

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   200m

I divide the Isosceles triangle

400 m2 - 100 m2 = Height m2

160,000 – 10,000 = 150,000

Height = square root of 150,000

Height = 387.29 to two decimal places

To calculate the area of this Isosceles triangle we calculate:

Area = ½ x base x height

Area = ½ x 200 x 387.29

Area = 38729 m2 (meter squared)

Triangle 4

I chose another size isosceles triangle to continue my investigation

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