An offshore oil well is located in the ocean at point $W$, which is 8 km from point $A$ on land as shown in the figure above. Oil is to be piped to point $B$, which is 12 km from point $A$, using a straight pipe laid underwater from $W$ to some point $P$ between $A$ and $B$, and then on to point $B$ via another straight pipe laid on land. If the cost of laying a pipe is $60 000 per km underwater and $45 000 per km on land, how far should point $P$ lie from point $A$ to minimise the cost of laying the pipe?

1. Formulate an equation for the total cost of laying the pipe based on the position of point $P$.

Let the distance $AP$ be $x$ km, where $0 \le x \le 12$.
By Pythagoras' theorem, the length of the underwater pipe is $WP = \sqrt{8^2 + x^2} = \sqrt{x^2 + 64}$ km.
The length of the pipe on land is $PB = 12 - x$ km.
The total cost $C$ (in $) is given by:
$C = 60000\sqrt{x^2 + 64} + 45000(12 - x)$

2. Differentiate the cost function with respect to $x$ to find the stationary point.

To find the minimum cost, we differentiate $C$ with respect to $x$ and set $\frac{dC}{dx} = 0$:
$\frac{dC}{dx} = 60000 \left( \frac{1}{2}(x^2 + 64)^{-\frac{1}{2}} (2x) \right) - 45000$
$\frac{dC}{dx} = \frac{60000x}{\sqrt{x^2 + 64}} - 45000 = 0$

3. Solve the algebraic equation to find the value of $x$.

$\frac{60000x}{\sqrt{x^2 + 64}} = 45000$
Dividing by 15000:
$\frac{4x}{\sqrt{x^2 + 64}} = 3$
$4x = 3\sqrt{x^2 + 64}$
Squaring both sides:
$16x^2 = 9(x^2 + 64)$
$16x^2 = 9x^2 + 576$
$7x^2 = 576$
Since distance $x > 0$,
$x = \sqrt{\frac{576}{7}} = \frac{24}{\sqrt{7}} \approx 9.0714 \text{ km}$

4. Verify that this stationary point results in a minimum cost.

Use the second derivative test:
$\frac{d^2C}{dx^2} = 60000 \left[ \frac{\sqrt{x^2+64} \cdot (1) - x \cdot \left(\frac{x}{\sqrt{x^2+64}}\right)}{x^2+64} \right]$
$\frac{d^2C}{dx^2} = 60000 \left[ \frac{x^2+64 - x^2}{(x^2+64)^{\frac{3}{2}}} \right] = \frac{3840000}{(x^2+64)^{\frac{3}{2}}}$
Since $\frac{d^2C}{dx^2} > 0$ for all valid $x > 0$, the cost $C$ is definitively minimised when $x = \frac{24}{\sqrt{7}} \approx 9.07$ km.