A drone is programmed to fly from its launch pad at the point \(P(1,0)\) along a path described by the equation \(y=\sqrt{x-1}\), \(x\geq1\), as shown in the figure above. The pilot of the drone is located at the point \(Q(10,0)\).
Show that the distance, \(L\), of the drone from the pilot as it travels along its flight path is given by \(\displaystyle L=\sqrt{x^2-19x+99}\).[1]
Use differentiation to find the exact minimum value of \(L\), proving that it is a minimum.[4]
The distance travelled by the drone along its flight path, denoted by \(F\), is given by the integral \[F(x)=\int_1^x\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2}\,\mathrm{d}x,\] when the drone is at the point \((x,y)\).
Given that the drone is flying at a constant speed of 0.8 units per second along its flight path, find the rate at which the shadow of the drone is moving horizontally when it is directly above its pilot. You may assume that the drone casts a shadow directly below it as it flies.[5]