1. Identify two examples of exponential r/s
2. The data below shows how the the population of the world has changed over the last three hundred years
Year population(mill)
1650 500
1700 600
1750 700
1800 900
1850 1300
1900 1700
1950 2500
2000 7000
The data can be modelled by a function of the form P=Poe^kt
where P is the pop in millions at t years and Po and K are constants;
a. What is the relevance of Po in the exp eqn
b. Transform the exp eqn to linear form
c. Use the linearised model to plot a graph and draw in the line of best fit
d. Find the eqn of the line of best fit. Use the eqn to find the values of K and Po and comment on your answers.
e. Use your model to predict the pop in year 2050
f. Use your model to estimate when the pop will be 10 000 million.
Po is the intial population: you can choose when. Let it be 1650.
7000=500 e^(350k)
solve for k (divide both sides by 500, then take the loge of each side).
To answer these questions, we need to understand exponential equations and how to analyze and manipulate data.
1. Examples of exponential growth or decay:
- Population growth: The example given in the question is a classic example of exponential growth where the population increases rapidly over time.
- Compound interest: Another example is the growth of a bank account with compound interest. The amount of money is increased exponentially based on the interest rate and the length of time.
2. Now, let's analyze the given data and answer the questions step by step:
a. The relevance of Po in the exponential equation is that it represents the initial population, or the population at the starting point (in this case, 1650). It helps determine the baseline for the population growth calculations.
b. To transform the exponential equation into linear form, we can take the natural logarithm (ln) of both sides of the equation. This helps in simplifying the equation and making it easier to analyze and graph.
Taking the natural logarithm of the equation P = Po * e^(kt) gives:
ln(P) = ln(Po) + kt
c. Using the linearized model, we can plot a graph with the natural logarithm of the population (ln(P)) on the y-axis and the years (t) on the x-axis. Then, we can draw the line of best fit to represent the trend in the data. This line will help us analyze and interpret the population growth pattern.
d. To find the equation of the line of best fit, we use statistical methods like linear regression analysis. This process determines the equation that best fits the data points (ln(P) and t). The equation will be in the form of y = mx + b, where y represents the vertical axis (ln(P)), x represents the horizontal axis (t), m is the slope of the line, and b is the y-intercept.
By performing linear regression analysis, we can obtain the equation of the line of best fit. The slope (k) and the y-intercept (ln(Po)) obtained from this equation can be used to determine the values of Po and k. Additionally, their values provide insights into the growth rate and initial population size.
e. To predict the population in the year 2050 using the obtained model, we substitute t=2050 into the linearized equation and solve for ln(P). Then, we can get the population (P) by taking the exponential function (e^x) of ln(P).
f. To estimate when the population will reach 10,000 million, we use the same approach as in the previous step. We substitute P = 10,000 million into the exponential equation (P = Po * e^(kt)), solve for t, and find the corresponding year.
Remember to perform the necessary calculations step by step and use appropriate software or tools for analyzing the data and generating the line of best fit.