Approaching Gelfand's Algebra problem 131 about a motorboat

[user10921]

Homework Statement

A motorboat needs a hours to go from A to B down the river and needs b hours to go from B to A(up the river). How many hours would it need to go from A to B if there were no current in the river?

Relevant Equations

$$vb_r question mark?$$

I am struggling on how to approach this problem. Hints would help greatly. This is also from Gelfand's Algebra problem 131

 

[berkeman]

The motorboat has an intrinsic speed with respect to the water. The water has a speed downstream. Those speeds add in the downstream direction and subtract in the upstream direction.

Define variables for those two speeds and write some equations describing what is going on. Please show that work. Thank you.

 

[user10921]

let $$v_r$$ be the river water speed and $$v_b$$ the boat speed and
let's let AB be the distance between A and B
when going down the river, the speed is $$v_b+v_r$$ and the duration is a, so the equation is
$$a=\dfrac{AB}{v_b+v_r}$$
similarly:
$$b=\dfrac{AB}{v_b-v_r}$$

 

[user10921]

What we're looking for is
$$x=\dfrac{AB}{v_b}$$
So we have to manipulate the equation to get rid of the $$v_r$$ out of the expression

 

[user10921]

So once you manipulate the equation, you get $$v_b+v_r=\dfrac{AB}{a}$$ and $$v_b-v_r=\dfrac{AB}{b}$$
Using system of equations you add the two and you get
$$2v_b=AB\left(\dfrac{1}{a}+\dfrac{1}{b}\right)$$
and divide by two on each side and you're left with that. Thank you for helping

 

[Mark44]

so the equation is $$a=\dfrac{AB}{v_b+v_r}$$

...and you get $$2v_b=AB\left(\dfrac{1}{a}+\dfrac{1}{b}\right)$$

These equations are a little confusing, as I originally thought that $AB$ was the product of A and B, the labels for the two points on the river. Instead, I think your intent is that $AB$ represents the distance from point A to point B. It would be clearer to define a new variable, say d, that represents the distance from A to B.

 

[aheight]

That doesn't tell you how long it takes.