If the speed of a boat in still water is a km/h and the speed of the water current is b km/h, then find the value of $$(a^2 + b^2)$$. It is given that the boat covers 3x km against the current in 0.5x hours and 2.4x km with the current in 0.3x hours, where x is a positive irrational quantity.

This is a classic question from boats and streams, where you need to apply the equation of 'speed = distance/time'.

Given that the speed of the boat in still water is a kmph, and the speed of the current is b kmph.

In the instance where the boat goes against the stream (resultant speed is a-b): 0.5x = $$\ \frac{\ 3x}{(a-b)}$$

From this, (a-b) = 6    ........(1)

In the instance where the boat goes along with the stream (resultant speed is a+b): 0.3x= $$\frac{\ 2.4x}{(a+b)}$$

From this, (a+b)= 8    ............(2)

Using the two equations: a=7 and b=1

Hence, the value of $$(a^2 + b^2)$$ is 50.