5. (14 marks) Suppose that each year, one fifth of all puppies become adult dogs

Question

5. (14 marks) Suppose that each year, one fifth of all puppies become adult dogs, while the other four fifths remain puppies. Suppose also that each year, two thirds of all adult dogs give birth to puppies (on average 5 puppies each) and one sixth of all adult dogs die. (a) (6 marks) Find a matrix relating the number of puppies and adult dogs this year to the number of puppies and adult dogs next year. to the number of puppies and adult dogs two years from now. and adult dogs were there last year? (Find all possible solutions). (b) (3 marks) Find a matrix relating the number of puppies and adult dogs this year (c) (5 marks) If this year there are 500 puppies and 650 adult dogs, how many puppies

Explanation / Answer

Here the key figures are :
4/5 of puppies remain puppies
2/3 of adult dogs give birth to avg 5 puppies (so 2/3*5 = 10/3)
1/5 of puppies become adul dogs
1/6 of adult dogs die so 5/6 of adult dogs live

so no: of adult dogs currently = a and number of adult dogs in next year = A
no: of puppies in current year =p and number of puppies in next year = P
A = 5/6 a + 1/5 p =p/5 + 5a/6
P = 4/5p + 2/3*a*5 = 4p/5 + 10a/3

So the matrix is:
[ P A] = [ p a ] [ 4/5 1/5
10/3 5/6 ]
Thus, the requried matrix is :[ 4/5 1/5
10/3 5/6 ]

B) For the second year, you will have to multiply the coefficient matrix with itself  

[ 4/5 1/5 * [ 4/5 1/5
10/3 5/6 ] 10/3 5/6 ]
=[ 16/25 + 10/15 4/25+5/30
40/15+50/18 10/15+25/36 ]

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