a tugboat goes 160 miles upstream in 20 hours. The return trip downstream takes 10 hours. Find the speed of the tugboat without a current and the speed of the current.
if the speed of the boat in still water is x, and the speed of the current is c, then, remembering that distance=speed*time,
20(x-c)=10(x+c)
20x-20c=10x+10c
10x=30c
x=3c
20(2c) = 160
c = 4
so, x=12
check:
20(8) = 160
10(16) = 160
To find the speed of the tugboat without a current and the speed of the current, we can set up a system of equations based on the given information.
Let's assume the speed of the tugboat without a current is represented by 'T' (in miles per hour) and the speed of the current is represented by 'C' (in miles per hour).
When the tugboat is going upstream (against the current), its effective speed is decreased. We can express this as:
T - C = Upstream speed
Similarly, when the tugboat is going downstream (with the current), its effective speed is increased. We can express this as:
T + C = Downstream speed
From the given information, we know that the tugboat travels 160 miles upstream in 20 hours. Using the formula Distance = Speed × Time, we can write:
T - C = 160/20
T - C = 8
Similarly, the tugboat travels the same distance downstream in 10 hours:
T + C = 160/10
T + C = 16
Now, we have two equations:
T - C = 8
T + C = 16
To solve these equations, we can use the method of adding the two equations together (eliminating C):
(T - C) + (T + C) = 8 + 16
2T = 24
T = 24/2
T = 12
So, the speed of the tugboat without a current is 12 miles per hour.
To find the speed of the current (C), we can substitute the value of T = 12 into one of the original equations, such as T + C = 16:
12 + C = 16
C = 16 - 12
C = 4
Therefore, the speed of the current is 4 miles per hour.