This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here is the solution to question 3: 3. Population Growth a) If the initial population is P, growth rate is R per annum, and population after T years is P_T, the formula to find P_T is: P_T = P (1 + (R)/(100))^T The formula is P_T = P (1 + (R)/(100))^T. b) After how many years will the population of the village be 20,808? Step 1: Identify the given values. Initial population, P = 20000 Annual growth rate, R = 2\% (from the problem description) Population after T years, P_T = 20808 Step 2: Use the population growth formula. P_T = P (1 + (R)/(100))^T Substitute the given values: 20808 = 20000 (1 + (2)/(100))^T 20808 = 20000 (1 + 0.02)^T 20808 = 20000 (1.02)^T Step 3: Solve for T. (20808)/(20000) = (1.02)^T 1.0404 = (1.02)^T Recognize that 1.0404 = (1.02)^2. (1.02)^2 = (1.02)^T Therefore, T = 2 years. The population will be 20,808 after 2 years. c) If the population increases at the rate of 3% per annum, by what number will the population of the village be increased in 2 years? Step 1: Identify the given values for this part. Initial population, P = 20000 New annual growth rate, R = 3\% Time, T = 2 years Step 2: Calculate the population after 2 years (P_T). P_T = P (1 + (R)/(100))^T Substitute the values: P_T = 20000 (1 + (3)/(100))^2 P_T = 20000 (1 + 0.03)^2 P_T = 20000 (1.03)^2 P_T = 20000 × 1.0609 P_T = 21218 Step 3: Calculate the increase in population. Increase = Final population - Initial population Increase = P_T - P Increase = 21218 - 20000 Increase = 1218 The population will be increased by 1218.