Exam-Style Question on Optimisation

The north corridor at Addwell High School has a width of \(2 \, \text{m}\). There is a ninety-degree corner at point \(C\). Points \(A\) and \(B\) are variable points on the base of the walls such that \(A\), \(C\), and \(B\) lie on a straight line.

Let \(L\) denote the length \(AB\) in metres.

Let \(\theta\) be the angle that \([AB]\) makes with the corridor wall, where \(0 < \theta < \frac{\pi}{2}\).

(a) Find the length of L in terms of \(\theta\).

(b) Find \(\frac{dL}{d\theta}\).

(c) When \(\frac{dL}{d\theta} = 0\), show that \(\theta = \frac{\pi}{4} \).

(d) Find \(\frac{d^2 L}{d\theta^2}\).

(e) Find \(\frac{d^2 L}{d\theta^2}\) when \(\theta = \frac{\pi}{4} \).

(f) Hence, justify that \(L\) is a minimum when \(\theta = \frac{\pi}{4} \).

(g) Determine this minimum value of \(L\).

Two people need to carry a pipe of length \(7 \, \text{m}\) along this corridor. The height of the corridor is \(2.5 \, \text{m}\).

(h) Determine whether this is possible, giving a reason for your answer.

(i) If the width of the pipe is considered to be negligible, what is theoretically the maximum length of a pipe that is able to be carried around the corner of the corridor?