An environmental scientist, Dr. Aris, believes the population of a certain species of fish in a lake, recorded as P P\,P thousands, has been growing exponentially since 2012. Dr. Aris models the population using the formula
P=a×bt P = a \times b^{t} P=a×btwhere t t\,t is the number of years since 2012 and a a\,a and b b\,b are constants.
He plots a graph of log10P\log_{10} Plog10P against t t\,t and determines the equation of the line of best fit to be:
log10P=0.041t+1.32 \log_{10} P = 0.041t + 1.32 log10P=0.041t+1.32(i) Show that, correct to three significant figures, a=20.9a = 20.9a=20.9.
(ii) Find the value of bbb, giving your answer to three significant figures.
According to the model, state the yearly percentage increase in the fish population.
(i) Use the model to predict the fish population in the year 2028.
(ii) Explain why the prediction made in part (c)(i) may be unreliable.
104 exam-style questions on OCR (MEI) A Level Maths 1.8 Exponentials and Logarithms, covering 1.8.1 The function y = a^x, 1.8.2 Convert between index and logarithmic form, 1.8.3 Logarithm as inverse of exponential, 1.8.4 Laws of logarithms, 1.8.5 Values of log_a a and log_a 1, 1.8.6 Solve equations of the form a^x = b, 1.8.7 Reduce y = ax^n and y = ab^x to linear form, 1.8.8 The function y = e^x, 1.8.9 Gradient of e^kx (A-level only), 1.8.10 The function y = ln x, and 1.8.11 Exponential growth and decay. Each one has a worked solution and a mark scheme showing where the marks go.