Question 34

An environmental biologist monitors the growth rate of a rare fern species. The rate of change of the leaf surface area, $R$ (in $cm2\text{cm}^2$ per week), is modeled by the function $\ln(t + 4)$ , where $t$ is the time in weeks after the initial measurement. The table below shows the observed values for the first 5 weeks.

$t$	0	1	2	3	4	5
$R (t)$	$ln⁡4\ln 4$	$ln⁡5\ln 5$	$ln⁡6\ln 6$	$ln⁡7\ln 7$	$ln⁡8\ln 8$	$ln⁡9\ln 9$

Using the trapezium rule with all the values in the table, show that the total change in surface area over the 5-week period can be approximated as

\int_{0}^{5} \ln(t + 4) \, dt \approx \ln k

where $k$ is an integer to be found.

Determine the value of $w$ that satisfies the equation

2\log_{4}(w + 3) - \log_{4}(w - 1) = 2

$t$	0	1	2	3	4	5
$R (t)$	$ln⁡4\ln 4$	$ln⁡5\ln 5$	$ln⁡6\ln 6$	$ln⁡7\ln 7$	$ln⁡8\ln 8$	$ln⁡9\ln 9$