To solve this problem, we need to understand the relationship between the speeds involved and the distances traveled by the boat downstream and upstream. Let's define the following variables:
- [tex]\( b \)[/tex] = speed of the boat in still water (km/h)
- [tex]\( c \)[/tex] = speed of the current (km/h)
- [tex]\( t_d \)[/tex] = time traveled downstream (hours) = 7 hours
- [tex]\( t_u \)[/tex] = time traveled upstream (hours) = 7 hours
- [tex]\( d_s \)[/tex] = distance from the starting point after 7 hours downstream and 7 hours upstream (km) = 154 km
When the boat is traveling downstream, the current aids its journey. Therefore, the effective speed downstream is [tex]\( b + c \)[/tex]. Conversely, when the boat travels upstream, it is going against the current, so the effective speed is [tex]\( b - c \)[/tex].
The distance traveled downstream:
[tex]\[ \text{Distance}_{\text{downstream}} = (b + c) \times t_d = (b + c) \times 7 \][/tex]
The distance traveled upstream:
[tex]\[ \text{Distance}_{\text{upstream}} = (b - c) \times t_u = (b - c) \times 7 \][/tex]
According to the problem, after 7 hours downstream and 7 hours upstream, the boat is 154 km away from its starting point. Therefore, the equation is:
[tex]\[ 7(b + c) - 7(b - c) = 154 \][/tex]
We simplify the equation:
[tex]\[ 7b + 7c - 7b + 7c = 154 \][/tex]
[tex]\[ 14c = 154 \][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[ c = \frac{154}{14} \][/tex]
[tex]\[ c = 11 \][/tex]
Thus, the speed of the current is [tex]\( 11 \)[/tex] km/h. Therefore, the correct answer is:
[tex]\[ \boxed{11 \text{ km/h}} \][/tex]
Among the provided options:
A) 20 km/h
B) 16 km/h
C) 11 km/h
D) 17 km/h
E) 13 km/h
The correct option is:
C) 11 km/h
