Motorboat Ques: Gelfand Algebra 131 | Hints to Help

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In summary: You need to find the time it takes to go from A to B. In summary, this problem is confusing and I need more help.
  • #1
user10921
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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
$$vb_r question mark?$$
Capture.JPG

I am struggling on how to approach this problem. Hints would help greatly. This is also from Gelfand's Algebra problem 131
 
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  • #2
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.
 
  • #3
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:
$$b=\dfrac{AB}{v_b-v_r}$$
 
  • #4
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
 
  • #5
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
 
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  • #6
user10921 said:
so the equation is $$a=\dfrac{AB}{v_b+v_r}$$
user10921 said:
...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.
 
  • #7
That doesn't tell you how long it takes.
 

Related to Motorboat Ques: Gelfand Algebra 131 | Hints to Help

1. What is Gelfand Algebra 131?

Gelfand Algebra 131 is a course in algebraic structures, named after the Russian mathematician Israel Gelfand, that covers topics such as group theory, ring theory, and field theory.

2. What is the difficulty level of Gelfand Algebra 131?

Gelfand Algebra 131 is considered to be an intermediate-level course, meaning that it is more advanced than introductory algebra courses, but not as advanced as graduate-level algebra courses.

3. What are some tips for succeeding in Gelfand Algebra 131?

Some tips for succeeding in Gelfand Algebra 131 include attending all lectures, actively engaging in class discussions, practicing problems regularly, and seeking help from the professor or a tutor when needed.

4. How can I prepare for Gelfand Algebra 131?

To prepare for Gelfand Algebra 131, it is recommended to have a strong foundation in basic algebra, as well as a good understanding of linear algebra and abstract algebra. It may also be helpful to review key concepts and definitions beforehand.

5. What career paths can Gelfand Algebra 131 lead to?

Gelfand Algebra 131 can lead to various career paths, including positions in mathematics, statistics, data analysis, and computer science. It can also be a solid foundation for graduate studies in mathematics or related fields.

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