O-Level Exponential and logarithmic functions: worked solution

Question

A radioactive sample’s mass y mg falls as time x hours passes, and an analyst proposes the model x²y = a + bx² with a and b constant. Explain how the readings can be turned into a straight-line graph, and how a and b would then be extracted from that line.

Model: x² y = a + b x². Divide the equation by x² is not needed; it is already linear if we treat X = x² and Y = x² y. Write Y = x² y = b x² + a, i.e. Y = b X + a where Y = x² y and X = x². So plot Y = x² y (vertical axis) against X = x² (horizontal axis). For each reading compute x² and x² y and plot the points; they should lie on a straight line. The gradient of the line equals b, and the vertical-axis intercept (where X = x² = 0) equals a.