Explanation:

Let the speed of speed of stream be u and speed of boat in still water be v.
Speed of boat in upstream = u - v
Speed of boat in downstream = u + v
A boat can go 3.6 km upstream and 5.4 km downstream in 54 minutes so,
Time = distance/speed
$$\frac{3.6}{u - v} + \frac{5.4}{u + v} = \frac{54}{60}$$
$$\frac{3.6}{u - v} + \frac{5.4}{u + v} = \frac{9}{10}$$
$$\frac{36}{u - v} + \frac{54}{u + v} = 9$$
4(u + v) + 6(u - v) = $$u^2 - v^2$$
10u - 2v = $$u^2 - v^2$$ ---(1)
Boat can go 5.4 km upstream and 3.6 km downstream in 58.5 minutes so,
$$\frac{5.4}{u - v} + \frac{3.6}{u + v} = \frac{58.5}{60}$$
$$\frac{54}{u - v} + \frac{36}{u + v} = \frac{585}{60}$$
$$\frac{6}{u - v} + \frac{4}{u + v} = \frac{13}{12}$$
72(u + v) + 48(u - v) = 13($$u^2 - v^2$$)
120u + 24v = 13($$u^2 - v^2$$)
Put the value of eq(1),
120u + 24v = 13 $$\times (10u - 2v)$$
120u + 24v = 130u - 26v
10u = 50v
u = 5v ---(2)
put the value of u in eq(1),
50v - 2v = $$25v^2 - v^2$$
48v = 24v^2
v = 2
put the value of v in eq(2),
u = 5$$\times$$ 2 = 10
Speed of boat in downstream = u + v = 10 + 2 = 12
The time taken by the boat in going 10 km downstream = distance/speed = 10/12 hours = 60 $$\times$$ 10/12 = 50 min