An atmospheric scientist monitors the concentration of a synthetic coolant gas, C C\,C parts per trillion (ppt), in the stratosphere. The scientist believes the concentration has been growing exponentially since the start of 2005 and proposes the model
C=p×qn C = p \times q^{n} C=p×qnwhere n n\,n is the number of years after 2005, and p p\,p and q q\,q are constants.
A graph of log10C\log_{10} Clog10C against n n\,n is plotted, and the resulting line of best fit has the equation:
log10C=0.052n+0.86 \log_{10} C = 0.052n + 0.86 log10C=0.052n+0.86(i) Show that, correct to three significant figures, p=7.24p = 7.24p=7.24.
(ii) Find the value of qqq, giving your answer to three significant figures.
Based on the model, determine the annual percentage increase in the concentration of the gas.
(i) Estimate the concentration of the gas at the start of 2025.
(ii) State one reason why this estimate might be inaccurate.
104 exam-style questions on OCR A Level Maths 1.6 Exponentials and Logarithms, covering 1.6.1 Properties of the exponential function, 1.6.2 Gradient of e^(kx) (A-level only), 1.6.3 Definition of the logarithm, 1.6.4 The natural logarithm function, 1.6.5 ln x as inverse of e^x, 1.6.6 Laws of logarithms, 1.6.7 Equations involving exponentials, 1.6.8 Reduction to linear form, and 1.6.9 Modelling using exponential functions. Each one has a worked solution and a mark scheme showing where the marks go.