Rectangular Area

mathdad

mathdad

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Beth has 300 feet of fencing available and wishes to enclose a rectangular area.

A. Express the area A of the rectangle as a function of x, where x is the length of the rectangle.

Let me try this one.
I will start with 300 = 2x + 2w
Note: Area = xw, where w is the width.

300 = 2x + 2w

300 - 2x = 2w

(300 - 2x)/2 = w

150 - x = w

A = xw

A = x(150 - x)

A = 150 - x^2

B. For what value of x is the area the largest? Note: x = -b/2a.

Let b = 150, and a = -1.

x = -b/2a

x = -(150)/2(-1)

x = -150/-2

x = 150/2

x = 75

C. What is the maximum area?

I need to evaluate the model found in part A when x = 75. Is this right? If I plug 75 into the model found in part A, I will get negative area. So, what did I do wrong?
 
You simply made a typo...you want:

[MATH]A(x)=150x-x^2[/MATH]
When optimizing a quadratic function, you can utilize the fact that the axis of symmetry lies midway between the two roots, which we can see by simple inspection must be \(x=75\).
 
I prefer to talk about half the perimeter and say that x+w = 150 instead of 2x + 2w = 300. This way those 2's are not in the way,
 
MarkFL said:
You simply made a typo...you want:

[MATH]A(x)=150x-x^2[/MATH]
When optimizing a quadratic function, you can utilize the fact that the axis of symmetry lies midway between the two roots, which we can see by simple inspection must be \(x=75\).
Click to expand...

A(x) = 150x - x^2

I dropped an x by mistake. Again, rushing through my computation. Mark, all my work is done on my cell phone. I do not have a PC or laptop.

Let x = 75

A(75) = 150(75) - (75)^2

A(75) = 11,250 - 5,625

A(75) = 5,625
 
mathdad said:
A(x) = 150x - x^2

I dropped an x by mistake. Again, rushing through my computation. Mark, all my work is done on my cell phone. I do not have a PC or laptop.

Let x = 75

A(75) = 150(75) - (75)^2

A(75) = 11,250 - 5,625

A(75) = 5,625
Click to expand...
Instead of multiplying those big numbers I would factor
A(x) = x(150-x)
A(75) = 75(150-75) = 75*75 = 5625.
You subtracted BIG numbers while I computed 150-75. You multiplied 150 with 75 and 75 with 75 while I just did (coincidentally) one of them namely 75*75
 
Jomo said:
Instead of multiplying those big numbers I would factor
A(x) = x(150-x)
A(75) = 75(150-75) = 75*75 = 5625.
You subtracted BIG numbers while I computed 150-75. You multiplied 150 with 75 and 75 with 75 while I just did (coincidentally) one of them namely 75*75
Click to expand...

I often forget how terribly important factoring is as a skill.
 
mathdad said:
My diagram makes no sense. I see it now.
Click to expand...

What I wanted you to recognize is that for a given fixed perimeter, the rectangle having that perimeter that encloses the greatest area is a square.
 
MarkFL said:
What I wanted you to recognize is that for a given fixed perimeter, the rectangle having that perimeter that encloses the greatest area is a square.
Click to expand...

Thank you, Mark. You have been my tutor and guide forever.
 
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