A medical researcher, Dr. Varma, monitors the concentration of a radioactive isotope, C C\,C mg/L, in a patient's bloodstream. She believes the concentration decays exponentially over time, where h h\,h is the number of hours after the initial injection. Dr. Varma models the concentration using the formula
C=p×qh C = p \times q^{h} C=p×qhwhere p p\,p and q q\,q are constants.
She plots a graph of log10C\log_{10} Clog10C against h h\,h and determines the equation of the line of best fit to be:
log10C=−0.065h+0.845 \log_{10} C = -0.065h + 0.845 log10C=−0.065h+0.845(i) Show that, correct to three significant figures, p=7.00p = 7.00p=7.00.
(ii) Find the value of qqq, giving your answer to three significant figures.
According to the model, state the hourly percentage decrease in the isotope concentration.
(i) Use the model to predict the isotope concentration 24 hours after the injection.
(ii) Explain why the prediction made in part (c)(i) may be unreliable.
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.