what's happening here? (printed area v. blue margins: my answer works, book's answer doesn't)

allegansveritatem

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132. A square page in a book measures 10 inches on each side. The printed matter on the page is to be surrounded by a blue margin of uniform width on all sides. The area of the blue margin is to be equal to the area of the portion of the page containing the printed matter. What are the dimensions of the portion of the page on which the print appears?



Here is how I set it up: 10^2 -(10-2x)^2= (10-2x)^2

Here is the answer: (I did this about 4 times by hand before using the calculator):

Solve [math]8c^2\, -\, 80c\, +\, 100\, =\, 0[/math] for [math]c[/math]
. . . . .[math]c\, =\, \dfrac{-5\,\left(\sqrt{2\,}\, -\, 2\right)}{2}\, \mbox{ or }\, c\, =\, \dfrac{5\, \left(\sqrt{2\,}\, +\, 2\right)}{2}[/math]
Here is the book's answer:

132. [math]5\, \sqrt{2\,}\, \mbox{in.}\, \times\, 5\, \sqrt{2\,}\, \mbox{in.; or }\, 7.1\, \mbox{in.}\, \times\, 7.1\, \mbox{in.}[/math]
My question is: Did the author or his representative do this problem the same way I did? I know my answer is right because it checks out.
 

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132. A square page in a book measures 10 inches on each side. The printed matter on the page is to be surrounded by a blue margin of uniform width on all sides. The area of the blue margin is to be equal to the area of the portion of the page containing the printed matter. What are the dimensions of the portion of the page on which the print appears?



Here is how I set it up: 10^2 -(10-2x)^2= (10-2x)^2

Here is the answer: (I did this about 4 times by hand before using the calculator):

Solve [math]8c^2\, -\, 80c\, +\, 100\, =\, 0[/math] for [math]c[/math]
. . . . .[math]c\, =\, \dfrac{-5\,\left(\sqrt{2\,}\, -\, 2\right)}{2}\, \mbox{ or }\, c\, =\, \dfrac{5\, \left(\sqrt{2\,}\, +\, 2\right)}{2}[/math]
Here is the book's answer:

132. [math]5\, \sqrt{2\,}\, \mbox{in.}\, \times\, 5\, \sqrt{2\,}\, \mbox{in.; or }\, 7.1\, \mbox{in.}\, \times\, 7.1\, \mbox{in.}[/math]
My question is: Did the author or his representative do this problem the same way I did? I know my answer is right because it checks out.
You haven't finished answering the question! What does your x mean? What are they asking for? How can you get the latter from the former?

Main lessons here: state the definition of your variables, and make sure you answer the question that is asked.

There is also a much quicker way to solve the problem, if you aim directly for what is asked.
 
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You haven't finished answering the question! What does your x mean? What are they asking for? How can you get the latter from the former?

Main lessons here: state the definition of your variables, and make sure you answer the question that is asked.

There is also a much quicker way to solve the problem, if you aim directly for what is asked.
Well, the x represents the width of the empty edge that borders the printed area of the page. I know they are asking for the length of the sides of the print square. I couldn't think of any other way to find this than by finding the width of the border. I will have to think about what you mean by aiming directly at what is asked for, namely the dimensions of the print square.
 
If the area of the print equals the area of the margin, then the area of the print is half of the total area ...

I presume you did determine that if the width x is -5(sqrt(2) - 2)/2, then the width of the print is 10 - 2*-5(sqrt(2) - 2)/2 = 5 sqrt(2), so that the book's answer agrees with yours. I can't say what method they used, as you asked; the end result doesn't reveal the method.
 
So actually we could just say that the answer is: the square root of 1/2(10)^2? Let's see: YES! square root of 50 is 5 times the square root of 2. Why didn't I think of that?
 
Partly, I imagine, because your mind was on algebra.

When we focus on a particular tool, we can miss other ways to do a task. That gives us more time practicing using that tool, but less time to spend on general problem solving skills, such as deciding what tool is best.

You aren't alone.
 
So actually we could just say that the answer is: the square root of 1/2(10)^2? Let's see: YES! square root of 50 is 5 times the square root of 2. Why didn't I think of that?
My advice is to always spend at least a few seconds on a problem to see if there is an obvious answer or maybe an easy way to solve it then the usual way.

Try to solve this one. (x+1)^2 = (x+1)
 
My advice is to always spend at least a few seconds on a problem to see if there is an obvious answer or maybe an easy way to solve it then the usual way.

Try to solve this one. (x+1)^2 = (x+1)
I will look at this a little later.
 
Partly, I imagine, because your mind was on algebra.

When we focus on a particular tool, we can miss other ways to do a task. That gives us more time practicing using that tool, but less time to spend on general problem solving skills, such as deciding what tool is best.

You aren't alone.
Exactly. I was concentrating on the means.
 
Yes, but how can that be solvable? (x+1)^2 can't = x+1. I mean, (x+1)^2 is x^2+2x+1.
0^2 = 0 and 1^2 = 1. What more do I need to say? If you do not understand then please write back
I agree that x^2+2x+1 is not identically x+1, but it is conditional on some values of x.
 
0^2 = 0 and 1^2 = 1. What more do I need to say? If you do not understand then please write back
I agree that x^2+2x+1 is not identically x+1, but it is conditional on some values of x.
if you set x to 0 then it makes sense. But solving this by the quadratic equation I get -1 plus or minus the square root of 1 over 4. Which, come to think of it is 0, -2/4. So yes, I guess you are right.
 
x^2+2x+1 = x+1
x^2+x=0
x(x+1) = 0
x=0 or x=-1

With quadratic formula: x= [-1 (+/-)sqrt(1-410)]/(2*1) = [-1 +/- 1]/2 = 0 or -1

Quick way: since 0^2 = 0 and 1^2 = 1 we have (x+1) = 0 or x+1 = 1. So x = 0 or -1
 
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