Appendix 2 · Mathematical Modelling

NCERT Class 10 Maths Exercise A2.2

All 3 questions of NCERT Class 10 Maths Exercise A2.2 (Mathematical Modelling), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   In each of the ​problems below, ​show the different stages of ​mathematical modelling for solving ​the problem. ​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.

   1. (i)What fraction of her second catch is ringed?
   2. (ii)Find an estimate of the total number of parrots in the field.
   Show answer

   (i) The fraction of the second catch that is ringed is $\frac{8}{40}=\frac{1}{5}$. (ii) Using $\frac{8}{40}=\frac{32}{N}$, the estimated total is $N=160$ parrots.

   Step-by-step solution →
2. Q2
   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.

   Show answer

   Estimate by sampling: cover the photograph with a grid of equal squares (here $6\times 4=24$), count the trees in a few squares chosen at random ($8$, $8$ and $7$), and multiply the average, $\frac{23}{3}$ trees a square, by the number of squares: $24\times \frac{23}{3}=184$, so about $184$ trees. This is the textbook's method: count the dots in a unit area and multiply by the total area.

   Step-by-step solution →
3. Q3
   A T.V. can be purchased for Rs 24000 cash or for Rs 8000 cashdown payment and six monthly instalments of Rs 2800 each. Ali goes to market to buy a T.V., and ​he has Rs 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?
   Show answer

   The instalment scheme is better. Following NCERT Example 2: balance after cash-down = 24000 - 8000 = 16000; instalments total 6 x 2800 = 16800, so interest = 800. Principal outstanding month by month = 16000, 13200, 10400, 7600, 4800, 2000, total 54000 rupee-months, so 54000 x r / (100 x 12) = 800 gives r = 17.78 % per annum (NCERT prints 17.74 %). That is less than the society’s 18 %, so the instalment scheme is cheaper.

   Step-by-step solution →