Answer:
The rate of the current is 2 mph.
Step-by-step explanation:
When the boat travels upstream we have:
[tex] v_{T_{1}} = v_{b} - v_{c} [/tex]
[tex] \frac{x_{1}}{t_{1}} = v_{b} - v_{c} [/tex]
Where:
[tex] v_{T}[/tex]: is the total speed
[tex]v_{b}[/tex]: is the speed of the boat = 10 mph
[tex]v_{c}[/tex]: is the speed of the current
x: is the distance
t: is the time
And when the boat travel downstream we have:
[tex]v_{T_{2}} = v_{b} + v_{c}[/tex]
[tex] \frac{x_{2}}{t_{2}} = v_{b} + v_{c} [/tex]
Since the boat takes 4 hours longer to travel 48 miles going upstream than it does to travel 24 miles going downstream we have:
[tex] \frac{x_{1}}{t + 4} = v_{b} - v_{c} [/tex] (1)
[tex] \frac{x_{2}}{t} = v_{b} + v_{c} [/tex] (2)
By adding equation (1) with (2):
[tex] \frac{x_{1}}{t + 4} + \frac{x_{2}}{t} = 2v_{b} [/tex]
[tex] tx_{1} + (t+4)x_{2} - 2v_{b}t(t+4) = 0 [/tex]
[tex] 48t + 24(t+4) - 2*10t(t+4) = 0 [/tex]
By solving the above equation for t we have:
t = 2 h
Now, by entering "t" into equation (2) we have:
[tex] \frac{24 miles}{2 h} = 10 mph + v_{c} [/tex]
[tex] v_{c} = 2 mph [/tex]
Therefore, the rate of the current is 2 mph.
I hope it helps you!
