Area as a function of x: A(x)=2xh. Where did the 2 come from?


by Wade2
Tags: graph, rectangle, rectangular plot
Wade2
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#1
Dec7-12, 12:21 PM
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This came from a friend of mine who has a math book on hand with a statement that doesn't seem to make sense. Neither of us are in school/college at the moment. This is math for the fun of it.

The area of the rectangular plot is the product of it's length (h) and width (x). We can write the area as a function of x: A(x)=2xh

Where did the 2 come from?
I've thought long and hard on this and even, at one point, ran into a reference to a barn and 100 feet of fence to section off a rectangular plot using the side of the barn.

I tested out A(x) = xh by itself and substituted h by making it relative to x (x/2 = h or that is to say that the height is 1/2 the width). When looked at it that way, the output for A(x) makes sense for the area. I cannot make sense of the 2 though in A(x) = 2xh.

So the question of the hour is as stated above.

Where did the 2 come from?
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micromass
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Dec7-12, 12:45 PM
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Which math book are you talking about?
Mark44
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Dec7-12, 01:07 PM
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The area of the rectangular plot is the product of it's length (h) and width (x). We can write the area as a function of x: A(x)=2xh
That formula is wrong. The area of a rectangle whose length is h and whose width is x is A = xh.

You might be misreading the information in the problem. If half the length is x, then the length is 2x, and then area would then be A = 2xh.

HomogenousCow
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Dec9-12, 11:36 AM
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Area as a function of x: A(x)=2xh. Where did the 2 come from?


Most like we're dealing with a function like y=3, when the author saids "area, he is probably referring to a rectangle extending equally on either side of the origin, hence A=2xh, since x is only half the extension.
arildno
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Dec9-12, 12:17 PM
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just a guess:

You have two sides, the overside and the underside, of the rectangular plot, each with area x*h
justsomeguy
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#6
Dec9-12, 12:31 PM
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This is from the CK-12 Trig book.

Example 6: You have 100 feet of fence with which to enclose a plot of land on the side of a barn. You
want the enclosed land to be a rectangle.

a. Write a function to model the area of the plot as a function of the width of the plot.

a. The equation: The area of the rectangular plot is the product of its length and width. We can write
the area as a function of x: A(x) = 2xh. We can eliminate h from the equation if we consider that we have
100 feet of fence, and we write an equation about how we are using that 100 feet of fence: x + 2h = 100.
(The fourth side of the rectangle does not require fence because of the barn.) We can solve this equation
for h and substitute into the area equation:

x + 2ft = 100 => 2h = 100 - x => h = 50 - -

A{x) = 2xh = 2x^50 -|)
A(x) = 100x-Jt 2
Mark44
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Dec9-12, 01:23 PM
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Quote Quote by justsomeguy View Post
This is from the CK-12 Trig book.
Example 6: You have 100 feet of fence with which to enclose a plot of land on the side of a barn. You
want the enclosed land to be a rectangle.

a. Write a function to model the area of the plot as a function of the width of the plot.

a. The equation: The area of the rectangular plot is the product of its length and width. We can write
the area as a function of x: A(x) = 2xh. We can eliminate h from the equation if we consider that we have
100 feet of fence, and we write an equation about how we are using that 100 feet of fence: x + 2h = 100.
(The fourth side of the rectangle does not require fence because of the barn.) We can solve this equation
for h and substitute into the area equation:

x + 2ft = 100 => 2h = 100 - x => h = 50 - -

A{x) = 2xh = 2x^50 -|)
A(x) = 100x-Jt 2
Either you're not copying what's in this book correctly or the book is unbelievably messed up.

I can begin to follow what they're doing, but then what you have here makes no sense.

We have 100 ft of fencing that will be used to fence in three sides of a rectangle.
From one equation they give -- x + 2h = 100 -- we can conclude that the side parallel to the barn has length x (ft) and the two sides perpendicular to barn each have length h (ft).

Skipping down to the end, you have
x + 2ft = 100 -- that should be x + 2h = 100

So 2h = 100 - x -- this is OK
Then you have h = 50 -- -- this is wrong

Next you have
A(x) = 2xh = 2x^50 -|)

and then
A(x) = 100x-Jt 2

Neither of these makes any sense.
justsomeguy
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#8
Dec9-12, 01:30 PM
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Don't shoot the messenger..

I copied it from here: http://archive.org/stream/ost-math-c...metry_djvu.txt

Didn't make any sense to me either.
phinds
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Dec9-12, 02:13 PM
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Quote Quote by justsomeguy View Post
Don't shoot the messenger..

I copied it from here: http://archive.org/stream/ost-math-c...metry_djvu.txt

Didn't make any sense to me either.
Copied as in "cut and paste"? (I'm not going to look through that whole thing to find what you are referencing.

If it was cut and paste, then why would you interpret the answer provided as "shooting the messenger"? Since the response was "if you have copied it correctly then the book is messed up". Why not respond, "thanks for confirming for me that the book is messed up"
justsomeguy
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Dec9-12, 02:17 PM
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[QUOTE=phinds;4190721]Copied as in "cut and paste"? (I'm not going to look through that whole thing to find what you are referencing.

You don't have to read the whole thing. CTRL-F, "A(x)" will take you right to it. I copied and pasted. The source I copied from may be flawed somehow.

Quote Quote by phinds View Post
If it was cut and paste, then why would you interpret the answer provided as "shooting the messenger"? Since the response was "if you have copied it correctly then the book is messed up". Why not respond, "thanks for confirming for me that the book is messed up"
Why I didn't say one thing instead of another.. I can't say, however.

Quote Quote by phinds View Post
Either you're not copying what's in this book correctly or the book is unbelievably messed up.
A 3rd possibility. The source I copied from is (believably) messed up.
Wade2
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#11
Dec10-12, 02:14 AM
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Thank you all for your efforts. I too found that example of the fence in a different website full of examples. My friend had the very same example out of her book that she tried to work. I also saw how you cannot have an answer of 50 for x since that means you have 50 on one side and the other and a big, open hole going out into the open.

Check Example 6 in the link below:

http://en.wikibooks.org/wiki/High_Sc...Lesson_Summary

This does appear to be flawed. If several brains like us can collaborate and see some fault here, then perhaps we are right.

I can always bring it up to someone tomorrow at the college nearby and see what the math lab has to say about it.
Curious3141
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Dec10-12, 02:40 AM
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Quote Quote by arildno View Post
just a guess:

You have two sides, the overside and the underside, of the rectangular plot, each with area x*h
Yes, because everyone plants crops on both sides of the plot.
arildno
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Dec10-12, 06:14 AM
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Quote Quote by Curious3141 View Post
Yes, because everyone plants crops on both sides of the plot.
Indeed.
If they are lichen-producers, it might actually work.
Michael Redei
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Dec10-12, 06:51 AM
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Quote Quote by Wade2 View Post
Check Example 6 in the link below:

http://en.wikibooks.org/wiki/High_Sc...Lesson_Summary

This does appear to be flawed. If several brains like us can collaborate and see some fault here, then perhaps we are right.
Maybe we just don't speak the same language. Here's an excerpt from that WikiBook, from the section entitled Vocabulary:
dependent variable
The input variable of a function, usually denoted x.
...
independent variable
The output variable of a function, usually denoted y.
This is the kind of thing that makes me mistrust any source whose name begins with "Wiki".


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