An environmental biologist monitors the growth rate of a rare fern species. The rate of change of the leaf surface area, RRR (in cm2\text{cm}^2cm2 per week), is modeled by the function R(t)=ln(t+4)R(t) = \ln(t + 4)R(t)=ln(t+4), where ttt is the time in weeks after the initial measurement. The table below shows the observed values for the first 5 weeks.
| ttt | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| R(t)R(t)R(t) | ln4\ln 4ln4 | ln5\ln 5ln5 | ln6\ln 6ln6 | ln7\ln 7ln7 | ln8\ln 8ln8 | ln9\ln 9ln9 |
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
∫05ln(t+4) dt≈lnk \int_{0}^{5} \ln(t + 4) \, dt \approx \ln k ∫05ln(t+4)dt≈lnkwhere kkk is an integer to be found.
Determine the value of www that satisfies the equation
2log4(w+3)−log4(w−1)=2 2\log_{4}(w + 3) - \log_{4}(w - 1) = 2 2log4(w+3)−log4(w−1)=2
