Because the added number is getting smaller every time, I think the total area will eventually stop. I will carry on for a few days to see if this is so.
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.
n=Day number (again, as the first day is not in the pattern, the day required must be represented by the number before it. E.g. For day 4, the sum is +256. This equals 592 – The area after 4 days of growth!).
This series is an infinite geometric series, this means that it will go on forever. However, as the terms are getting smaller each time, eventually, they will get so small that they will make no significant difference to the total area. This has been mentioned before. In this particular example, the total area will never increase beyond 597 cm2. I found a formula that tells us the sum to infinity of this series. It uses the same substitutes and it is , therefore in this example, .
This completes my investigation. However, I have decided to extend it to see if the formulas work on triangles as well.
Here are the first three days of growth in an equilateral Martian triangle…
Day 1
Height = = = Area = x
=
Day 2
Height =
=
=
Area = x
=
Total Area = + 3()
=
=
Day 3
Height =
=
=
Area = x
=
Total Area =
=
=
I am going to put these results in a table to make it look a bit neater.
Again, if I don’t include the first day in the pattern, I can use the following substitutes for the formula …
a=3()
r=
n=day number (again, the day number is represented by the number before it, e.g. day 3 is represented by 2.)
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.
