Mathematical Modelling

Consider the following situation.

A problem dating back to the early 13th century, posed by Leonardo Fibonacci asks how many rabbits you would have if you started with just two and let them reproduce. Assume that a pair of rabbits produces a pair of offspring each month and that each pair of rabbits produces their first offspring at the age of 2 months. Month by month the number of pairs of rabbits is given by the sum of the rabbits in the two preceding months, except for the 0th and the 1st months.

After just 16 months, you have nearly 1600 pairs of rabbits !

Clearly state the problem and the different stages of mathematical modelling in this situation.

Answer

From table,

F₀ = 1

F₁ = 1

F₂ = F₀ + F₁

where the nos. represent pair of rabbits.

So, after n months, pairs of rabbits can be represented by

F_n = F_{n - 1} + F_{n - 2}

An ornithologist wants to estimate the number of parrots in a large field. She uses a net to catch some, and catches 32 parrots, which she rings and sets free. The following week she manages to net 40 parrots, of which 8 are ringed.

(i) What fraction of her second catch is ringed?

(ii) Find an estimate of the total number of parrots in the field.

Answer

(i) Given,

Ornithologist manages to net 40 parrots, of which 8 are ringed.

Fraction of ringed = $\dfrac{8}{40} = \dfrac{1}{5}$.

Hence, $\dfrac{1}{5}$ of her second catch is ringed.

(ii) We may apply the idea of marking and recapture to determine the overall number of parrots in the field. According to the theory, the percentage of marked parrots in the second catch corresponds to the percentage of marked parrots in the overall population.

Let's use N to represent the overall number of parrots in the field. The first batch, as said, caught 32 parrots and ringed every single one of them. In the second batch, there were 8 ringed parrots out of 40 caught. We can set up a proportion :

$\Rightarrow \dfrac{32}{N} = \dfrac{8}{40} \\[1em] \Rightarrow N = \dfrac{32 \times 40}{8} \\[1em] \Rightarrow N = 32 \times 5 \\[1em] \Rightarrow N = 160.$

Hence, total number of parrots in the field are 160.

Suppose the adjoining figure represents an aerial photograph of a forest with each dot representing a tree. Your purpose is to find the number of trees there are on this tract of land as part of an environmental census.

Answer

Steps :

1. Take 1 cm² area and count no. of trees (let it be n).

2. Calculate area of forest (let it be a).

3. Multiply total area and no. of trees in 1 cm² area.

Hence, total number of trees in forest will be an.

A T.V. can be purchased for ₹ 24000 cash or for ₹ 8000 cash down payment and six monthly instalments of ₹ 2800 each. Ali goes to market to buy a T.V., and he has ₹ 8000 with him. He has now two options. One is to buy TV under instalment scheme or to make cash payment by taking loan from some financial society. The society charges simple interest at the rate of 18% per annum simple interest. Which option is better for Ali?

Answer

In installment scheme :

Cash down payment = ₹ 8000

Each monthly installment = ₹ 2800

So, total installment = ₹ 2800 × 6 = ₹ 16800.

So, total T.V. cost = ₹ 16800 + ₹ 8000 = ₹ 24800.

In loan system :

Cash down payment = ₹ 8000

Remaining amount = ₹ 24000 - ₹ 8000 = ₹ 16000

Now, ₹ 16000 at 18% per annum for 6 months or 0.5 year.

S.I. = $\dfrac{16000 \times 18 \times 0.5}{100}$ = ₹ 1440.

So, total T.V. cost = ₹ 8000 + ₹ 16000 + ₹ 1440 = ₹ 25440.

Since, total cost is more in loan system.

Hence, six monthly installments scheme is better for Ali.

Based upon the data of the past five years, try and forecast the average percentage of marks in Mathematics that your school would obtain in the Class X board examination at the end of the year.

Answer

For past five years average Mathematics marks in a school are :

2019 = 90%

2020 = 95%

2021 = 92%

2022 = 93%

2023 = 91%

Steps of construction :

1. Plot the percentage on y-axis and years on x-axis.
2. Plot the points.
3. Construct the best fit line.
4. For estimation of percentage in 2024, draw a straight line parallel to y-axis from point I = 2024 touching best fit line at point F from F draw a line parallel to x-axis touching y-axis at point J.

From graph,

J = 95.

Hence, at the end of the year marks obtained in class X mathematics will be 95%.