Maths population question

Question

The number of kangaroos in a farm is causing some concern. Scientists sent to investigate the problem find that, on their arrival, there are 51 000 kangaroos on the farm.

a) The kangaroos population on the farm increases at a constant rate of 1000 kangaroos per week.

1) How many kangaroos will be on the farm one week after scientists arrival?

2) Write down a rule for the no. of kangaroos on the farm n weeks after the arrival of the scientists.

3) How many weeks would it take for the kangaroo population to grow to 70 000?

4) Calculate the constant weekly increase that would lead to a kangaroo population of 68 5000 on this farm, 35 weeks after the scientists arrival. (i think something like tn= t (n-1)d)

Answer 1

1) 52000 kangaroos

2) 51000 + (no. weeks X 1000)

3) 19000 kangaroos

4) UNSURE (this is not my answer, I just couldn't work it out)

Good luck, sorry about not being able to work #4 out, but I hope the rest are right.

Cheers bud

Answer 2

1. 51,000 + 1,0000 = 52,000

2.  If we let w be the number of weeks which have elapsed since the scientists' arrival, then the number of kangaroos is given by this equation:

k = 51,000 + 1,000w.

3.  Plug the numbers into the equation above:

70,000 = 51,000 + 1,000w

19,000 = 1,000w

19 = w

4.  The constant difference from week to week is 1,000.  Since this is an arithmetic progression, then we can use this formula:

a(n) = a(1) + (n - 1) d, where a(n) is the population after n weeks, a(1) is the initial population, n is the number of elapsed weeks, and d is the constant rate of growth of the population per week.

Here we have to think a bit.  The standard formula for the nth term of an arithmetic progression is a(n) = a(1) + (n - 1) d.  If we let n = k + 1, where k is the number of elapsed weeks since the initial population was observed, then a(n) =  a(k + 1) = a(1) + [(k + 1) - 1] = a(1) + (k) d.  So, if we plug the appropriate numbers into the last equation, we get this:

a(n) = a(1) + [(k + 1) - 1] d

a(n) = a(1) + (k) d

68,500 = 51,000 + (35) d

17,500 = 35 d

500 = d.

So, for a kangaroo population of 68,500 on week 35, the population must grow by 500 kangaroos each week.