Mathema cs Por olio work
Estimation of the population of a particular region/country under the assumptions
that there is no migration in or out of the existing population in a particular year.
Task to be done
1. Find the population of a selected region in a particular year.
2. Find the number of births and number of deaths in the existing population in a
particular year 𝑡 (say). Let
𝑃(𝑡): denote the population in a particular year 𝑡.
𝐵(𝑡): denote the number of births in one year between 𝑡 and 𝑡 + 1.
𝐷(𝑡): denote the number of deaths in one year between 𝑡 and 𝑡 + 1.
3. Obtain the relation
𝑃(𝑡 + 1) = 𝑃(𝑡) + 𝐵(𝑡) − 𝐷(𝑡) ……………….(1)
4. Assume that
( )
𝑏= ( )
represents the birth rate for the time interval 𝑡 to 𝑡 + 1.
( )
𝑑= ( )
represents the death rate for the time interval 𝑡 to 𝑡 + 1.
5. From (1), we have
𝑃(𝑡 + 1) = 𝑃(𝑡) + 𝐵(𝑡) − 𝐷(𝑡)
( ) ( )
= 𝑃(𝑡) 1 + ( )
− ( )
= 𝑃(𝑡)(1 + 𝑏 − 𝑑) ……………….(2)
6. Taking 𝑡 = 0 in equation (2), we get
𝑃(1) = 𝑃(0)(1 + 𝑏 − 𝑑).
For 𝑡 = 1, we get
𝑃(2) = 𝑃(0)(1 + 𝑏 − 𝑑)
Continuing above equation, we get
𝑃(𝑡) = 𝑃(0)(1 + 𝑏 − 𝑑) ……………….(3)
Here it is assumed that birth rate and death rate remains the same for
consecutive years. 𝑃(0) denote the initial population. Equation (3) gives the
mathematical model for calculating the population in 𝑡 year.
7. Using calculator find the population in different number of years.
8. Compare the population data obtained theoretically and draw the inferences.
OR
To minimise the cost of the food, meeting the dietary requirements of the staple
food of the adolescent students of your school.
Task to be done
1. Make a survey of at least 100 students to find which staple food they consume
on daily basis.
2. Select two food items constituting one cereal and one pulse.
3. Find from dietician the minimum requirement of protein and carbohydrate for an
adolescent and also find the content of protein and carbohydrate in 1 kg. of
selected cereal and pulse respectively.
4. Find the minimum cost of the selected cereal and pulse from market.
5. Formulate the corresponding Linear Programming problem.
6. Solve the problem graphically.
7. Interpret the result.
OR
Formation of differential equation to explain the process of cooling of boiled
water to a given room temperature.
Task to be done
1. Boil 1 litre of water in a pan/beaker.
2. Note the room temperature and the temperature of the boiled water.
3. Note the temperature at an interval of every half hour till the temperature of the
water reaches the room temperature. Prepare a corresponding table as shown
below:
Time (𝑡) at an Temperature of Room
interval of hour water (𝑇) Temperature (𝑃) Difference 𝑇 − 𝑃
4. Let 𝑇 denote the temperature of the boiled water at time 𝑡. 𝑃 denote the room
temperature under the assumption it remains constant throughout the experiment
∝ 𝑇 − 𝑃 or = −𝑘(𝑇 − 𝑃), 𝑘 is proportionality constant and minus sign signifies
that temperature is decreasing.
or = −𝑘 𝑑𝑡. Integrating, we get
log|𝑇 − 𝑃| = −𝑘𝑡 + 𝐶 ………………..(1)
5. Find the value of 𝐶 and 𝑘 using two initial values of 𝑇 and 𝑡 from the observation
table to get the particular solution of the differential equation (1).
