Kilbaha 2016 Exam 2

[Hydroxyl]

Just a question from this exam for Recursion and Financial Modelling:

Question 6 (4 marks)
The committee decides to increase the number of gum trees on the island. Mr. Smith, a committee member, suggests that they increase the number of gum trees by 4% of the number of gum trees on the island at the end of each year. It is observed that 12 trees are destroyed each year by natural disasters. Mr. Smith writes the equation Tn+1 = aTn + b T0 = 86 to model this situation.
a. What are the values of a and b in this equation?  a = 1.04, b = -12
c. Mrs Nguyen suggests that they should increase the number of gum trees on the island by 20% instead of 4%.
After how many years will the number of gum trees first be greater than 200?
My answer: Recurrence Relation is now T0 = 86, Tn+1= 1.2Tn-12. Therefore, After 10 Years, the Value will be greater than 200 (T10)

Now what I'm confused about is in the answers, They did:
Tn+1 =1.2^n ×86−12>200
Use calculator to find this occurs when n = 5.
After 5 years.

If the number of gum trees is increasing by 20%, with 12 trees destroyed each year, shouldn't the new recurrence relationship be T0 = 86, Tn+1= 1.2Tn-12?
This means that the year in which the number of trees are greater than 200 would be 10. The sequence generated from this relation is 86, 91.2, 97.44, 104.928, 113.914, 124.696. In the answers, it has the same recurrence relation but they only subtracted 12 once, despite saying that 12 trees are destroyed EACH YEAR. So, 12 should be subtracted from the previous value at each step, and not just at the end.


Would be appreciated if someone can clarify this for me!


Thanks

[clarke54321]

Just a question from this exam for Recursion and Financial Modelling:

Question 6 (4 marks)
The committee decides to increase the number of gum trees on the island. Mr. Smith, a committee member, suggests that they increase the number of gum trees by 4% of the number of gum trees on the island at the end of each year. It is observed that 12 trees are destroyed each year by natural disasters. Mr. Smith writes the equation Tn+1 = aTn + b T0 = 86 to model this situation.
a. What are the values of a and b in this equation?  a = 1.04, b = -12
c. Mrs Nguyen suggests that they should increase the number of gum trees on the island by 20% instead of 4%.
After how many years will the number of gum trees first be greater than 200?
My answer: Recurrence Relation is now T0 = 86, Tn+1= 1.2Tn-12. Therefore, After 10 Years, the Value will be greater than 200 (T10)

Now what I'm confused about is in the answers, They did:
Tn+1 =1.2^n ×86−12>200
Use calculator to find this occurs when n = 5.
After 5 years.

If the number of gum trees is increasing by 20%, with 12 trees destroyed each year, shouldn't the new recurrence relationship be T0 = 86, Tn+1= 1.2Tn-12?
This means that the year in which the number of trees are greater than 200 would be 10. The sequence generated from this relation is 86, 91.2, 97.44, 104.928, 113.914, 124.696. In the answers, it has the same recurrence relation but they only subtracted 12 once, despite saying that 12 trees are destroyed EACH YEAR. So, 12 should be subtracted from the previous value at each step, and not just at the end.


Would be appreciated if someone can clarify this for me!


Thanks

Hey Hydroxyl,

Yes, you're right.  It should be 10 years rather than 5 years.

Just to be sure I put this situation into my financial solver (even though it's not a financial situation) and found that my N value was 9.2339...., which must be rounded up to 10 (10 years).

The equation used in the answers only takes into account the 12 trees being destroyed once, as opposed to every year. 

[Hydroxyl]

Hey Hydroxyl,

Yes, you're right.  It should be 10 years rather than 5 years.

Just to be sure I put this situation into my financial solver (even though it's not a financial situation) and found that my N value was 9.2339...., which must be rounded up to 10 (10 years).

The equation used in the answers only takes into account the 12 trees being destroyed once, as opposed to every year. 

Thanks Clarke!