The population of a medium size town has been growing at a rate of (about) % per year since 1-1-2000. There were inhabitants on 1-1-2000.
Set up a formula to calculate the number of inhabitants as a function of time in years since 1-1-2000.
Draw the corresponding graph on semi-logarithmic paper.
On this graph, find the number of inhabitants on 1-1-2015. Check your answer using the formula.
The table below shows measurements made in a bacterial culture. is given in hours, and is the number of bacteria.
t | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
N | 50 | 84 | 141 | 237 | 398 | 670 | 1125 |
Use these measurements to make another table showing against .
Draw the corresponding graph. Can you approximate the graph with a straight line? Is this exponential growth?
Write down a formula of as a function of .
Using your answer in c, write down a formula for as a function of .
The graph of an increasing amount as a function of time is shown here plotted on semi-logarithmic paper.
Set up a formula for .
Calculate, rounded to two decimals, the value of where . Use the graph to check your answer.
Somewhere in the negative values of , the graph has an intersect with the -axis. Calculate the corresponding value of , rounded to two decimals.
Mammals change from trot to gallop when they reach a certain threshold pace (the number of steps per minute). The pace at which this happens appears to depend on the body weight (in kg). Take (in kg) to be the body weight and to be the pace. The straight line goes through the points belonging to small dogs and horses.
How can you see that a logarithmic scale has been used on both axes?
Since a logarithmic scale has been used on both axes, the plot actually shows the relationship of with . The point for horses then represents and . Determine yourself the (approximate) values for the point belonging to the small dog.
Now derive a formula for as a function of .
Using the properties of logarithms you can now derive a formula for as a function of . Show all steps in your derivation.