Referring to the graph below,
- Find the growth factor between the numbers of stick insects in consecutive time intervals of two weeks.
- Hence find a relationship between x (time in days) and y (number of insects) that can be used as an approximate model for the data given in the graph.
I worked out that the two weekly growth factor is $\approx1.5$ - so that's all fine. I just don't know how to go about doing part ii.
The textbook says that the answer to ii) is $y=6.7(15^\frac{x}{14})$. Now I'm quite confused and not really sure how to arrive at that answer from the graph. I see that its in the form $y=ab^x$ but not sure where they get the $6.7$ or $15$ from.
$\endgroup$1 Answer
$\begingroup$Either the textbook has a printing mistake or you have a reading or typing mistake. A good answer to the problem is
$$6.7(1.5^{x/14})$$
Notice the decimal point in the $1.5$ that is not in the answer that you typed.
Here is an explanation.
First you need to find the growth factor by dividing consecutive $y$-values and taking an average. I get a value of $1.44$, but your $1.5$ is pretty good and is in fact an excellent growth factor for weeks up through $10$. (The later weeks bring the average growth factor down.)
A formula for geometric growth is
$$y=P(r^{x/t})$$
where $y$ is the final population, $P$ is the beginning population when $x=0$, $r$ is the growth rate between two times that are $t$ units apart, and $x$ is the time. In other words, we sample the data at times $t$ apart and want the formula to use times that are $1$ apart.
We now need to find $P$, the beginning population when $x=0$. Note that there is no point on your graph for time zero, so we have to calculate it. We can take the growth backwards from the second week to the zero-th week by dividing the population at the second week by the growth rate. So,
$$P=\frac{10}{1.5}\approx 6.7$$
Our data was sampled two weeks apart, which is $14$ days apart, and we want our final formula to use time in days, so $t=14$. We already know that $r=1.5$.
Substitute all those values in the general formula, and we get the textbook's answer.
This can be checked by drawing our own graph. This matches the first five points quite well. Note that I changed the $x$-axis values to days rather than weeks, to match the formula we are checking.
If we use my overall growth rate of $1.44$, we get an initial population of about $6.9$ and get this graph, which is a better match for the last three points.
