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Find all the solutions to the this equation.


Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

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Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Whose Line Graph Is it Anyway?

Age 16 to 18
Challenge Level


Scientific processes involving two variables can often be represented using equations and line graphs.

In this problem, $9$ processes, their equations and graphs have been mixed up and shown below. In each case, the two variables are represented by the letters $x$ and $y$ and the labels from the axes of the graphs have been removed.

Which can you match up? What is the interpretation of the variables $x$ and $y$ in each case?

Can you identify the physical interpretation of three key points on each of the graphs?



  1. Number of rapidly dividing bacteria present in a food-limited environment, starting from a small initial sample.
  2. Concentration in the blood of a drug following an injection.
  3. Angle of oscillation of a real pendulum of length $1$m in air.
  4. Volume (litres) against pressure (atmospheres) for $1$ mole of an ideal gas at $0^\circ$ C.
  5. Vertical distance travelled by a small, heavy ball dropped from a plane.
  6. Rate of reaction of a catalysed reaction in terms of the concentration of reagent.
  7. Number of rapidly dividing bacteria present in a food-rich environment, starting from a small initial sample.
  8. Hours of daylight per day in a town in the far northern hemisphere.
  9. Model of the distance of the Earth from the sun in astronomical units.

Line Graphs


A: $y(x) = 4.9 x^2$



B: $y(x) =500 \times 2^{-0.6667x}$



C: $y(x) =1- 0.01671\cos(0.0172 x)$



D: $y(x) = 12+10\sin(0.121 x)$



E: $y(x) = 5\cos(3.13 x)e^{-0.05x}$



F: $y(x) = (11.3 x)/(2.1+x)$



G: $y(x) =10 \times 2^{4x}$



H: $y(x) = \frac{1000000}{10+(100000-10)2^{-4x}}$



I: $y(x)x = 22.4133$




The numbers have been carefully chosen to represent certain time/length/unit scales for particular physical phenomena. Can you deduce the reason for the choice of any of the numbers?