Solve the problem. A cruise boat travels 48 miles downstream in 3 hours and returns upstream in 6 hours. Find the rate of the stream.
Downstream: 48=(vb+vs)3
upstsream: 46=(vb-vs)6
multiply the first equation by 2
1) 96=6vb+6vs
2) 46=6vb-6vs
subtract the equations:
50=12vs solve for vs.
i don't understand the way this is being worded. where did 46 come from? what does vs and vb stand for?
There is a mistake in bobpursley's equations - both should have the same distance of 48.
vs = velocity of stream
vb = velocity of boat
and Distance = Velocity*Time
Downstream
48 = (vb + vs)*3
The stream velocity is +ve as the time is shorter downstream - the stream helps
So vb + vs = 16
Upstream
48 = (vb - vs)*6
The stream velocity is -ve as the time is longer so the stream hinders the boat
So vb - vs = 8
Then just solve the 2 equations simultaneously to get vs = 4
Check that and also find the velocity of the boat.
I hope that helps.
To solve this problem, we need to use the concept of relative speed. Let's denote the speed of the boat as B and the speed of the stream as S.
When the boat is traveling downstream, i.e., in the same direction as the current, the effective speed of the boat will be increased by the speed of the stream. Therefore, the speed of the boat relative to the ground will be B + S.
Conversely, when the boat is traveling upstream, i.e., in the opposite direction to the current, the effective speed of the boat will be reduced by the speed of the stream. So, the speed of the boat relative to the ground will be B - S.
We are given that the boat travels 48 miles downstream in 3 hours. Using the formula distance = speed × time, we have:
48 = (B + S) × 3
Similarly, when the boat travels upstream, it takes 6 hours to cover the same distance of 48 miles. Using the same formula, we get:
48 = (B - S) × 6
We now have a system of two equations with two unknowns. Let's solve it:
From the first equation, we can rewrite it as B + S = 48/3 = 16.
From the second equation, we can rewrite it as B - S = 48/6 = 8.
Now, we can solve these two equations simultaneously. Add the two equations together:
(B + S) + (B - S) = 16 + 8
2B = 24
B = 24/2
B = 12
Substituting this value of B back into one of the original equations, we get:
12 + S = 16
S = 16 - 12
S = 4
Therefore, the rate of the stream is 4 miles per hour.
So, the rate of the stream is 4 miles per hour.