how does this work?

[allegansveritatem]

Here is problem:


Here is my solution:


and here is solutions book solution:


I really can't follow this solution from the authors. I get the sense that they are forming a quadatic equation out of the data but I just can't follow this past the first line. I get the feeling that this is a high teck way of doing what I did. I got the approximate and the books gives the exact. Can someone unpack this book solution for me. I wonder if I need to put aside this precalculus book and maybe find a book that is a kind of preprecalculus book?

 

[lev888]

7500x is not correct. Instead of x you need the length of the diagonal segment that's crossing the river. See the triangle? One side is x, another is 1. Can you calculate the third side?

 

[Denis]

What exactly is your problem/question?

We start with:
7500sqrt(x^2 + 1) + 6000(5 - x) = 35000
Divide by 500:
15sqrt(x^2 + 1) + 12(5 - x) = 70
Re-arrange:
15sqrt(x^2 + 1) = 70 - 12(5 - x)
Simplify:
15sqrt(x^2 + 1) = 12x + 10
Square both sides:
225(x^2 + 1) = 144x^2 + 240x + 100
Continue:
225x^2 + 225 = 144x^2 + 240x + 100
81x^2 - 240x + 125 = 0

Solve for x and you'll get same as "the book's answer"!!

 

[allegansveritatem]

7500x is not correct. Instead of x you need the length of the diagonal segment that's crossing the river. See the triangle? One side is x, another is 1. Can you calculate the third side?

Well, here is what I was thinking of:
View attachment 11709

I estimated that the square root of the hyponeuse squared was pretty close to 5. But the above image is exactly what I had in mind. But, I really want to know what the **** the solutions manual is doing.

 

[allegansveritatem]

What exactly is your problem/question?

We start with:
7500sqrt(x^2 + 1) + 6000(5 - x) = 35000
Divide by 500:
15sqrt(x^2 + 1) + 12(5 - x) = 70
Re-arrange:
15sqrt(x^2 + 1) = 70 - 12(5 - x)
Simplify:
15sqrt(x^2 + 1) = 12x + 10
Square both sides:
225(x^2 + 1) = 144x^2 + 240x + 100
Continue:
225x^2 + 225 = 144x^2 + 240x + 100
81x^2 - 240x + 125 = 0

Solve for x and you'll get same as "the book's answer"!!

well, where does the x^2 plus 1 come from exactly. Where does the 15 come from? Where does the 15 sqrt(x^2 +1) + 12(5-x) come from? HOW do you get that from the first equation and I ask again, HOW do you even get the first equation? In my last post I didn't use any squared expressions and why should I have? I mean, I knew already that the distance, as the crow flies, between the power station and the town khad to be 1 plus the square of 5. And so I used the sqrt of 26 and x as the basis for my variable.

 

[dealing]

Honestly I would just chill and listen to music or something. Math is so much work.

 

[dealing]

well, where does the x^2 plus 1 come from exactly. Where does the 15 come from? Where does the 15 sqrt(x^2 +1) + 12(5-x) come from? HOW do you get that from the first equation and I ask again, HOW do you even get the first equation? In my last post I didn't use any squared expressions and why should I have? I mean, I knew already that the distance, as the crow flies, between the power station and the town khad to be 1 plus the square of 5. And so I used the sqrt of 26 and x as the basis for my variable.

I sort of agree, but this is not necessary.

 

[pka]

Here is problem:
View attachment 11706

Here is my solution:
View attachment 11707

and here is solutions book solution:
View attachment 11708

I really can't follow this solution from the authors. I get the sense that they are forming a quadatic equation out of the data but I just can't follow this past the first line. I get the feeling that this is a high teck way of doing what I did. I got the approximate and the books gives the exact. Can someone unpack this book solution for me. I wonder if I need to put aside this precalculus book and maybe find a book that is a kind of preprecalculus book?

Have Look THIS LINK.

 

[allegansveritatem]

I'm not sure that I was able to upload this image so I will upload it again. This what I was thinking about with the solution I posted at the head of this thread. I just estimated the distance as 5 but here it is made clearer what I was thinking about:


Why is this wrong? And, what is going on with the solution from the book.

 

[allegansveritatem]

What exactly is your problem/question?

We start with:
7500sqrt(x^2 + 1) + 6000(5 - x) = 35000
Divide by 500:
15sqrt(x^2 + 1) + 12(5 - x) = 70
Re-arrange:
15sqrt(x^2 + 1) = 70 - 12(5 - x)
Simplify:
15sqrt(x^2 + 1) = 12x + 10
Square both sides:
225(x^2 + 1) = 144x^2 + 240x + 100
Continue:
225x^2 + 225 = 144x^2 + 240x + 100
81x^2 - 240x + 125 = 0

Solve for x and you'll get same as "the book's answer"!!

This just a copy of what is already in the solutions manual. Please, where does the variable come from? Where does the 15 come from and the 12 etc.

 

[Steven G]

This just a copy of what is already in the solutions manual. Please, where does the variable come from? Where does the 15 come from and the 12 etc.

When you have a worked out solution you may have to think about what is going on. I saw 75000, 6000 and 35000; now I see 15, 12 and 70. How can that be? Ah, they divided each number by 500. OK??

As far as where does the variable come from--it was in the diagram.

 

[MarkFL]

well, where does the x^2 plus 1 come from exactly. Where does the 15 come from? Where does the 15 sqrt(x^2 +1) + 12(5-x) come from? HOW do you get that from the first equation and I ask again, HOW do you even get the first equation? In my last post I didn't use any squared expressions and why should I have? I mean, I knew already that the distance, as the crow flies, between the power station and the town khad to be 1 plus the square of 5. And so I used the sqrt of 26 and x as the basis for my variable.

In the diagram, you see that the length of the underwater pipe is the hypotenuse of a right triangle having legs \(x\) and \(1\), and so by Pythagoras, the hypotenuse is:

[MATH]\sqrt{x^2+1}[/MATH]
Now, if we consider that the total cost is the cost per mile for underwater pipe times the number of miles underwater plus the cost per mile overland times the number of miles overland, this leads to the equation:

[MATH]7500\sqrt{x^2+1}+6000(5-x)=35000[/MATH]
And the solution then follows as given by your book. When you get into Calc I, you'll be able to determine the value of \(x\) that minimizes the expense of the pipeline.

 

[lev888]

Well, here is what I was thinking of:
View attachment 11709

I estimated that the square root of the hyponeuse squared was pretty close to 5. But the above image is exactly what I had in mind. But, I really want to know what the **** the solutions manual is doing.

In a right triangle with legs x and 1 what is the length of the hypotenuse?

 

[Denis]

well, where does the x^2 plus 1 come from exactly. Where does the 15 come from?
Where does the 15 sqrt(x^2 +1) + 12(5-x) come from?
HOW do you get that from the first equation and I ask again,
HOW do you even get the first equation?

Did you bother to LOOK at the diagram:

A

1

D    x    C          5-x            B

AC is underwater (@$7500), CB is overland (@$6000).

ADC forms a right triangle, so:
AC = sqrt(1^2 + x^2) = sqrt(x^2 + 1)
so:
7500sqrt(x^2 + 1) + 6000(5 - x) = 35000

Are you finally "ready to rumble"?

 

[allegansveritatem]

When you have a worked out solution you may have to think about what is going on. I saw 75000, 6000 and 35000; now I see 15, 12 and 70. How can that be? Ah, they divided each number by 500. OK??

As far as where does the variable come from--it was in the diagram.

well, I didn't see the divisor anywhere, so these numbers seemed to sprout out of thin air. I would advise the author of this solutions manual not to write any Math for Dummies books.

 

[allegansveritatem]

In the diagram, you see that the length of the underwater pipe is the hypotenuse of a right triangle having legs \(x\) and \(1\), and so by Pythagoras, the hypotenuse is:

[MATH]\sqrt{x^2+1}[/MATH]
Now, if we consider that the total cost is the cost per mile for underwater pipe times the number of miles underwater plus the cost per mile overland times the number of miles overland, this leads to the equation:

[MATH]7500\sqrt{x^2+1}+6000(5-x)=35000[/MATH]
And the solution then follows as given by your book. When you get into Calc I, you'll be able to determine the value of \(x\) that minimizes the expense of the pipeline.

I see that now. I think part of my problem is I misread the diagram. I didn't really understand what the triangle signified. I thought it was there as a kind of suggestion that this is one way to do the job. But it seems to be saying this is THE way to do it.

 

[allegansveritatem]

Did you bother to LOOK at the diagram:

A

1

D    x    C          5-x            B

AC is underwater (@$7500), CB is overland (@$6000).

ADC forms a right triangle, so:
AC = sqrt(1^2 + x^2) = sqrt(x^2 + 1)
so:
7500sqrt(x^2 + 1) + 6000(5 - x) = 35000

Are you finally "ready to rumble"?

I think I can work it out now...but I will have to wait til the morning when my brain comes back to life.

 

[allegansveritatem]

When you have a worked out solution you may have to think about what is going on. I saw 75000, 6000 and 35000; now I see 15, 12 and 70. How can that be? Ah, they divided each number by 500. OK??

As far as where does the variable come from--it was in the diagram.

Part of my problem is I didn't interpret the diagram in the way it was intended to be interpreted. At this late hour I am not sure exactly what I thought that triangle was hinting at.

 

[allegansveritatem]

Thanks to everyone who contributed to this thread. I think I can work out the solution now myself...but I would still like to know why the solution that I posted, I mean the solution suggested by the photo of the calculator screen, why that idea is not a correct solution. Or is it? It is correct but just not correct enough? It was good enough for rock and roll but not Mozart?

 

[MarkFL]

Thanks to everyone who contributed to this thread. I think I can work out the solution now myself...but I would still like to know why the solution that I posted, I mean the solution suggested by the photo of the calculator screen, why that idea is not a correct solution. Or is it? It is correct but just not correct enough? It was good enough for rock and roll but not Mozart?

You can approximate \(\sqrt{x^2+1}\) with \(x\), and as \(x\) gets larger, the better the approximation it will be. Think of a right triangle with a vertical leg fixed at one unit, and the horizontal leg is free to change. If this variable leg is short, then this leg and the hypotenuse will be noticeably different in length, but as you let the leg grow in length, this leg and the hypotenuse become more and more similar in length, relative to one another.