Applied matrix modeling problems

6 At the beginning of the year John had a total of \(\pounds 2000\) in three different accounts. He has twice as much money in the current account as in the savings account.

• The current account has an interest rate of \(2.5 \%\) per annum.
• The savings account has an interest rate of \(3.7 \%\) per annum.
• The supersaver account has an interest rate of \(4.9 \%\) per annum.

John has predicted that he will earn a total interest of \(\pounds 92\) by the end of the year.

1. Model this situation as a matrix equation.
2. Find the amount that John had in each account at the beginning of the year.
3. In fact, the interest John will receive is \(\pounds 92\) to the nearest pound. Explain how this affects the calculations.

1. The population of chimpanzees in a particular country consists of juveniles and adults. Juvenile chimpanzees do not reproduce.

In a study, the numbers of juvenile and adult chimpanzees were estimated at the start of each year. A model for the population satisfies the matrix system $$\binom { J _ { n + 1 } } { A _ { n + 1 } } = \left( \begin{array} { c c } a & 0.15 \\ 0.08 & 0.82 \end{array} \right) \binom { J _ { n } } { A _ { n } } \quad n = 0,1,2 , \ldots$$ where \(a\) is a constant, and \(J _ { n }\) and \(A _ { n }\) are the respective numbers of juvenile and adult chimpanzees \(n\) years after the start of the study.

1. Interpret the meaning of the constant \(a\) in the context of the model. At the start of the study, the total number of chimpanzees in the country was estimated to be 64000 According to the model, after one year the number of juvenile chimpanzees is 15360 and the number of adult chimpanzees is 43008
2. 1. Find, in terms of \(a\) $$\left( \begin{array} { c c } a & 0.15 \\ 0.08 & 0.82 \end{array} \right) ^ { - 1 }$$
   2. Hence, or otherwise, find the value of \(a\).
   3. Calculate the change in the number of juvenile chimpanzees in the first year of the study, according to this model. Given that the number of juvenile chimpanzees is known to be in decline in the country,
3. comment on the short-term suitability of this model. A study of the population revealed that adult chimpanzees stop reproducing at the age of 40 years.
4. Refine the matrix system for the model to reflect this information, giving a reason for your answer.
   (There is no need to estimate any unknown values for the refined model, but any known values should be made clear.)

1. In this question you must show all stages of your working.

A college offers only three courses: Construction, Design and Hospitality. Each student enrols on just one of these courses. In 2019, there was a total of 1110 students at this college.
There were 370 more students enrolled on Construction than Hospitality.
In 2020 the number of students enrolled on

• Construction increased by \(1.25 \%\)
• Design increased by \(2.5 \%\)
• Hospitality decreased by \(2 \%\)

In 2020, the total number of students at the college increased by \(0.27 \%\) to 2 significant figures.

1. 1. Define, for each course, a variable for the number of students enrolled on that course in 2019.
   2. Using your variables from part (a)(i), write down three equations that model this situation.
2. By forming and solving a matrix equation, determine how many students were enrolled on each of the three courses in 2019.