An ecological study tracked the expansion of an invasive water lily population in a lake starting on 1 March. The total surface area covered by the lilies, S S\,S square metres, was recorded over several months. The area is modelled by the equation
S=abt S = ab^t S=abtwhere t t\,t is the number of months since 1 March, and a a\,a and b b\,b are constants.
Show that S=abtS = ab^tS=abt can be written in the form log10S=log10a+tlog10b\log_{10} S = \log_{10} a + t \log_{10} blog10S=log10a+tlog10b.
The values of log10S\log_{10} Slog10S against t t\,t are plotted on a graph using data collected for 0≤t≤80 \leq t \leq 80≤t≤8, and a line of best fit is drawn. The line passes through the points (0,1.48)(0, 1.48)(0,1.48) and (8,3.72)(8, 3.72)(8,3.72). Find estimates for the values of a a\,a and bbb. Give your answers to three significant figures.
Use the model to predict the month in which the surface area covered by the lilies will first exceed 25,000 m2m^2m2.
State one reason why the prediction in part (c) may be unreliable.