how does this work?

allegansveritatem said:
It is correct but just not correct enough? It was good enough for rock and roll but not Mozart?
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It might be good enough for an engineer, but not for a mathematician! In a math problem, unless you are told to approximate, you shouldn't, because the problems are given to allow you to practice the math, not just to solve a practical problem.

Actually, it probably wouldn't be good enough for an engineer, because they care about money. See if you can figure out how many dollars your route would cost, and compare that to $35,000! (On the other hand, the problem is not at all realistic, because the engineer would want to minimize the cost, not just to meet an arbitrary budget.)

Because the distances are not very large compared to 1 mile, your estimate necessarily is not very exact. See MarkFL's answer.
 
Mark, a question:
B
1
C.........................................................infinity..............................................................>A

Does that mean hypotenuse AB > infinity ? :)

AB = sqrt(1^2 + (infinity)^2)
 
Denis said:
Mark, a question:
B
1
C.........................................................infinity..............................................................>A

Does that mean hypotenuse AB > infinity ? :)

AB = sqrt(1^2 + (infinity)^2)
Click to expand...
Nope!!

Because

Infinity + 1 = Infinity + 2
 
Denis said:
Mark, a question:
B
1
C.........................................................infinity..............................................................>A

Does that mean hypotenuse AB > infinity ? :)

AB = sqrt(1^2 + (infinity)^2)
Click to expand...

[MATH]\lim_{x\to\infty}\frac{\sqrt{x^2+1}}{x}=\lim_{x\to\infty}\sqrt{1+\frac{1}{x^2}}=1[/MATH]
 
allegansveritatem said:
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.
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The issue here, in my opinion, is that each extra line in a book cost money and the publisher wants to save money even though it cost students lots of extra time to understand something.

However, hopefully in the end you learned a bit about reading math with steps being skipped!
 
MarkFL said:
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. :)
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I will have to think about this until it clarifies for me. Thanks
 
Jomo said:
The issue here, in my opinion, is that each extra line in a book cost money and the publisher wants to save money even though it cost students lots of extra time to understand something.

However, hopefully in the end you learned a bit about reading math with steps being skipped!
Click to expand...
Yes I certainly did that. Maybe it's for the best.
 
I went at this problem again today. I started out confident I now knew what was going on and that I would make short work of it. Hubris. Ten minutes later I had crashed into a wall of complexity. It wasn't that I didn't understand the problem...I didn't understand the solution. so I went back to the solutions manual and began another round of pondering it. I was able to follow the author's procedure until I got to 15 times the sqrt of x^2 +1 = 12x +10. I puzzled over this a few minutes and then suddenly it dawned on me that the author had performed operations on one side of the = sign and then transferred the results to the other side and had expected me to know what he'd done without having to show me. Silly of him. Anyway, once I saw this the rest was easy.
What follows is a rough version of how I would present this solution if I were writing a solutions book, a book, by the way, that I would title: Stepped Down Solutions.

11716
 
Looks good to be but at the end you did NOT show where the 27 in the denominator came from!
 
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