a tugboat goes 160 miles upstream in 20 hours. The return trip downstream takes 5 hours. Find the speed of the tugboat without a current and the speed of the current.
if tugboats speed is s, and current is c,
since the distances are the same, and
distance = speed * time,
(s-c)*20 = (s+c)*5 = 160
20s - 20c = 160
5s + 5c = 160
or,
20s+20c = 640
40s = 800
s = 20
c = 12
check:
160/32 = 5
160/8 = 20
To find the speed of the tugboat without a current and the speed of the current, we can use the concept of relative speed. Let's assume the speed of the tugboat without a current is "x" miles per hour and the speed of the current is "y" miles per hour.
When the tugboat travels upstream against the current, its effective speed reduces since it has to work against the current. The distance covered upstream is 160 miles, and it takes 20 hours. So, the effective speed of the tugboat going upstream is:
Speed upstream = Speed of the tugboat without current - Speed of the current
160 = (x - y) * 20
Similarly, when the tugboat travels downstream with the current, its effective speed increases since it gets assistance from the current. The distance covered downstream is still 160 miles, but it takes only 5 hours. So, the effective speed of the tugboat going downstream is:
Speed downstream = Speed of the tugboat without current + Speed of the current
160 = (x + y) * 5
Now, we have a system of two equations with two variables. We can solve these equations simultaneously to find the values of "x" (speed without current) and "y" (speed of the current).
160 = (x - y) * 20 ... Equation 1
160 = (x + y) * 5 ... Equation 2
First, let's simplify Equation 1 by dividing both sides by 20:
8 = x - y
Similarly, let's simplify Equation 2 by dividing both sides by 5:
32 = x + y
We can now solve these simplified equations simultaneously to find the values of "x" and "y".
Adding the two equations together eliminates the "y" variable:
8 + 32 = x - y + x + y
40 = 2x
Dividing both sides by 2:
20 = x
So, the speed of the tugboat without a current is 20 miles per hour.
Now that we have the value of "x", we can substitute it into either Equation 1 or Equation 2 to find the value of "y". Let's use Equation 1:
8 = (20 - y)
Subtracting 20 from both sides:
-12 = -y
Dividing both sides by -1:
12 = y
So, the speed of the current is 12 miles per hour.
Therefore, the speed of the tugboat without a current is 20 miles per hour, and the speed of the current is 12 miles per hour.