# A Martian Mushroom starts off as a square.

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Introduction

Page

## STATEMENT OF PROBLEM

A Martian Mushroom starts off as a square.

The next day it grows so that a new square half the length of the original square is added to each side.

Then the next day each of these squares increases by a new square half the length of yesterdays.

I am going to investigate the formula for the growth in area of a martian mushroom. I will begin by drawing the diagrams for the first four days of growth.

Day 1

Area = 162

Area = 256cm2

Day 2

Area = 162 + 4(82)

Area = 256 + 256

Area = 512cm2

Day 3

Area = 162 + 4(82) + 4(42)

Area = 256 + 256 + 64 Area = 576cm2

Day 4

Area = 162 + 4(82) + 4(42) + 4(22) Area = 256 + 256 + 64 + 16 Area = 592cm2

I am now going to have a look at the areas to see if there is a pattern to them.

Day | Sum of Area | Total Area |

1 | 256 | 256 |

2 | 256 + 256 | 512 |

3 | 256 + 256 + 64 | 576 |

4 | 256 + 256 +64 + 16 | 592 |

I have noticed that each time you add on of the previous number.

Middle

597.3333282

15

256 + 256 + 64 + 16 + 4 + 1 + + + 1/64 + 1/256 + 1/1024 + 1/4096 + 1/16384 + 1/65536 + 1/262144

597.3333321

16

256 + 256 + 64 + 16 + 4 + 1 + + + 1/64 + 1/256 + 1/1024 + 1/4096 + 1/16384 + 1/65536 + 1/262144 + 1/1048576

597.3333331

17

256 + 256 + 64 + 16 + 4 + 1 + + + 1/64 + 1/256 + 1/1024 + 1/4096 + 1/16384 + 1/65536 + 1/262144 + 1/1048576 + 1/4194304

597.3333333

18

256 + 256 + 64 + 16 + 4 + 1 + + + 1/64 + 1/256 + 1/1024 + 1/4096 + 1/16384 + 1/65536 + 1/262144 + 1/1048576 + 1/4194304 + 1/16777216

597.3333333

The maximum surface area for a martian mushroom that starts off as 16cm by 16cm is 597cm2

From this information I can investigate the formula for martian mushrooms.

I looked in the ‘A’ level book “Pure Mathematics 1” and found some formulas that work for this pattern…

To find the sum of the areas for a particular day, I found the formula

. Because in my investigation I have included the first term where it is not in the pattern, I will also have to add on 256, so the formula is actually +256. Here are the substitutes…

a=Area of original shape i.e. the added area on day 2, because day 1 is not in the pattern.

r=ratio. I mentioned previously that I am adding on a quarter of the original area each time. This therefore, is the ratio.

Conclusion

As before, I will have to add to the end of this formula to make it work in this series.

I will now try to prove that the formula works…

+

This should equal the total area of the Martian mushroom after 3 days.

I will work this out to see if it is so…

Here I have worked out the brackets, and started to solve the top half of the left side of the equation.

= I have solved the top half of the left side of the equation.

= Here I have multiplied the right hand side by 192, to get 768 as the common denominator.

= I have now added the two sides together.

= This is the simplified equation.

After the equation has been worked out and simplified, it equals , which is what I got before! This is proof that the equation does work for any geometric series of this kind, and the end of my coursework.

Dale Caffull

10 - 4

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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