# Speed of Current

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##### Full Member
The speed of a boat in still water is 10 mph. It travels 24 miles upstream and 24 miles downstream in 5 hours. What is the speed of the current?

Solution:

Of course this involves D = rt.

Upstream (boat):

D = 24 miles
rate = 10 mph

Downstream ( boat):

D = 24 miles
rate = ?

Total time of trip 5 hours.

What is the equation set up for this problem? Can I say we have a system of linear equations in two variables?

#### MarkFL

##### Super Moderator
Staff member
I would let $$c$$ be the speed of the current. Then we may write:

Upstream:

$$\displaystyle 24=(10-c)t$$

Downstream:

$$\displaystyle 24=(10+c)(5-t)$$

I would solve the first equation for $$t$$, and substitute this into the second equation, then you have an equation in $$c$$ only, which you can solve.

#### Otis

##### Senior Member
Sullivan may not have explained the physics.

Have you ever walked on a pedestrian conveyor belt? Your speed increases because the belt carries you along, as you walk.

net rate = [walking speed] + [belt speed]

If you walk on the belt the wrong way (opposite direction), then your speed decreases because the belt carries you backward, as you struggle to walk forward.

net rate = [walking speed] - [belt speed]

The situation is the same for airplanes flying with or against the wind, as well as for boats moving with or against the current.

'Upstream' means against the current (river water flows downstream). Therefore, the upstream rate is:

[boat speed] - [current]

because the water slows down the boat, by pushing against it.

The downstream rate is:

[boat speed] + [current]

because the water speeds up the boat, by carrying it along.

##### Full Member
Sullivan may not have explained the physics.

Have you ever walked on a pedestrian conveyor belt? Your speed increases because the belt carries you along, as you walk.

net rate = [walking speed] + [belt speed]

If you walked on the belt the wrong way (opposite direction), then your speed decreases because the belt carries you backward, as you struggle to walk forward.

net rate = [walking speed] - [belt speed]

The situation is the same for airplanes flying with or against the wind, as well as for boats moving with or against the current.

'Upstream' means against the current (river water flows downstream). Therefore, the upstream rate is:

[boat speed] - [current]

because the water slows down the boat, by pushing against it.

The downstream rate is:

[boat speed] + [current]

because the water speeds up the boat, by carrying it along.

Great work here!

##### Full Member
I would let $$c$$ be the speed of the current. Then we may write:

Upstream:

$$\displaystyle 24=(10-c)t$$

Downstream:

$$\displaystyle 24=(10+c)(5-t)$$

I would solve the first equation for $$t$$, and substitute this into the second equation, then you have an equation in $$c$$ only, which you can solve.
Exactly my thoughts--TWO EQUATIONS IN TWO VARIABLES.

##### Full Member
I would let $$c$$ be the speed of the current. Then we may write:

Upstream:

$$\displaystyle 24=(10-c)t$$

Downstream:

$$\displaystyle 24=(10+c)(5-t)$$

I would solve the first equation for $$t$$, and substitute this into the second equation, then you have an equation in $$c$$ only, which you can solve.
For the upstream equation, I can solve for t to get t = 24/(10 - c). I then replace t in the downstream equation with t = 24/(10 - c) and solve for c, the current.

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