Direct and Inverse Proportion

Two identical lamps are \(12\) m apart. A sensor lies on the straight line between them, \(x\) metres from Lamp \(A\).

Each lamp produces \(80\) lux at a distance of \(3\) m. Each obeys an inverse-square model, and their illuminations add.

1. Form an expression for total illumination at the sensor.

2. Calculate the total illumination when \(x=3\) and when \(x=6\).

3. Explain why doubling the sensor’s distance from Lamp \(A\) from \(3\) m to \(6\) m does not divide the total illumination by \(4\).

4. Find another sensor position with the same total illumination as at \(x=3\), without solving a polynomial equation.