A tugboat goes 180 miles upstream in 10 hours. The return trip downstream takes 5 hours. Find the speed of the tugboat without a current and the speed of the current.
speed of boat --- x mph
speed of current -- y mph
10(x-y) = 180 ---> x-y = 18
5(x+y) = 180 ---> x+y = 36
add them
2x = 54
x = 27
then y = 9
state proper conclusion
To solve this problem, we can use the concept of relative speed. Let's assume the speed of the tugboat in still water is T miles per hour and the speed of the current is C miles per hour.
When the tugboat goes upstream, it is moving against the current. So the effective speed is reduced by the speed of the current, i.e., (T - C) miles per hour.
On the other hand, when the tugboat goes downstream, it is moving with the current. So the effective speed is increased by the speed of the current, i.e., (T + C) miles per hour.
Using the formula Distance = Speed × Time, we can write two equations:
Equation 1: (T - C) × 10 = 180 (upstream journey)
Equation 2: (T + C) × 5 = 180 (downstream journey)
Let's solve these equations to find the values of T and C.
From Equation 1:
T - C = 18 (Dividing both sides by 10)
From Equation 2:
T + C = 36 (Dividing both sides by 5)
Now, we can solve these two equations simultaneously to get the values of T and C.
Adding both equations together:
(T - C) + (T + C) = 18 + 36
Simplifying:
2T = 54
Dividing both sides by 2:
T = 27
Substituting the value of T into one of the equations, let's use Equation 1:
27 - C = 18
Simplifying:
C = 9
So, the speed of the tugboat without a current is 27 miles per hour, and the speed of the current is 9 miles per hour.