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Emily F.

asked • 01/21/15

area of 4700 ft. ^2. Its perimeter is 400 ft. Write the area as a function of the length of the rectangle. Find the values of the length and width.

Please help, I want to understand how to answer it, not just the answer

2 Answers By Expert Tutors

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Brett H. answered • 01/21/15

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Emily F.

I'm sorry, I don't completely understand. I know how to do what you're saying, but it's not turning out right. Would you do a step by step with the actual numbers please? I'm trying to make sure I know how to do these. Thank you in advance for all the trouble.
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01/21/15

Brett H.

No problem! So, the first thing we want to do is solve for L.
L*W=A
L=A/W
L=4700/W
 
Then we solve for W in our perimeter equation.
2L+2W=400
2W=400-2L
W=200-L
 
We then want to substitute in what we found for W in our first equation.
L=4700/W
L=4700/(200-L)
 
Then we want to get all the L's on the same side.
L=4700/(200-L)
(200-L)*L=4700
 
Then we distribute.
(200-L)*L=4700
200L-L2=4700
 
Then we see that it is going to be a quadratic equation, so we will need everything on one side, set equal to 0.
200L-L2=4700
-L2+200L=4700
-L2+200L-4700=0
 
Now since this isn't a "nice" quadratic equation with easy, factorable numbers, we are definitely going to want to use the quadratic formula, just to make things easier, with the quadratic formula being:
x=(-b+√b2-4ac)/2a
or
x=(-b-√b2-4ac)/2a
The only difference between the two is the plus sign and the minus sign after the first b.
Our equation is set up in the form of ax2+bx+c, so now we just plug in the values of a, b and c into our quadratic formula:
L=(-200+√2002-4*-1*-4700)/2*-1
or
L=(-200-√2002-4*-1*-4700)/2*-1
So if we simplify those down, we get:
L=(-200+√21200)/-2
L=100+((√21200)/2)
For a decimal approximation we get L=27.1989
or
L=(-200-√21200)/-2
L=100-((√21200)/2)
For a decimal approximation we get L=172.8011
 
In this instance both of these answers are correct because L can equal both, because the value of L is interchangeable with W, so if we said L=27.1989 ft, then W=172.8011 ft. Or L=172.8011 ft and W=27.1989 ft
 
I hope that clears things up for you!
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01/21/15

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