4.3 Q-17

Question Statement

The population of Pakistan in 1961 was 60 million, and in 1981 it was 95 million. Using $t$ as the number of years after 1961, find the equation of the line that represents the population in terms of $t$. Then, use this equation to find the population in:

• (a) 1947
• (b) 1997

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Background and Explanation

This problem requires using the equation of a straight line to model the population growth of Pakistan over time. The key steps are:

1. Linear Equation: We will use the formula for the equation of a line, $P - P_1 = m(t - t_1)$, where:

   • $P$ is the population at time $t$,
   • $t$ is the number of years since 1961,
   • $P_1$ and $t_1$ are the population and year at one known point, and
   • $m$ is the slope of the line.

2. Slope Calculation: The slope $m$ can be calculated as the change in population divided by the change in time:
   $m = \frac{P_2 - P_1}{t_2 - t_1}$

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Solution

1. Identify the Known Points:
   From the given data, we know:

   • $P_1 = 60$ million in 1961, so $t_1 = 1961$,
   • $P_2 = 95$ million in 1981, so $t_2 = 1981$.

   Thus, the points are $(t_1, P_1) = (1961, 60)$ and $(t_2, P_2) = (1981, 95)$.

2. Calculate the Slope $m$:
   Using the slope formula: $m = \frac{P_2 - P_1}{t_2 - t_1} = \frac{95 - 60}{1981 - 1961} = \frac{35}{20} = \frac{7}{4}$
   So, the slope of the line is $m = \frac{7}{4}$.

3. Write the Equation of the Line:
   Using the point-slope form of the equation of a line $P - P_1 = m(t - t_1)$ and substituting $P_1 = 60$, $m = \frac{7}{4}$, and $t_1 = 1961$, we get: $P - 60 = \frac{7}{4}(t - 1961)$
   Expanding this: $P = 60 + \frac{7}{4}(t - 1961)$

4. Find the Population for 1947 (a):
   To find the population in 1947, substitute $t = 1947$ into the equation: $P = 60 + \frac{7}{4}(1947 - 1961)$
   Simplifying the terms: $P = 60 + \frac{7}{4}(-14) = 60 - 49 = 11 \text{ million}$
   Therefore, the population in 1947 was 11 million.

5. Find the Population for 1997 (b):
   To find the population in 1997, substitute $t = 1997$ into the equation: $P = 60 + \frac{7}{4}(1997 - 1961)$
   Simplifying the terms: $P = 60 + \frac{7}{4}(36) = 60 + 63 = 123 \text{ million}$
   Therefore, the population in 1997 was 123 million.

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Key Formulas or Methods Used

• Slope Formula:
  $m = \frac{P_2 - P_1}{t_2 - t_1}$

• Equation of a Line:
  $P - P_1 = m(t - t_1)$

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Summary of Steps

1. Identify the two points: $(1961, 60)$ and $(1981, 95)$.
2. Calculate the slope $m$ using the formula.
3. Write the equation of the line: $P = 60 + \frac{7}{4}(t - 1961)$.
4. To find the population in 1947, substitute $t = 1947$ into the equation: $P = 11$ million.
5. To find the population in 1997, substitute $t = 1997$ into the equation: $P = 123$ million.