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# maximum

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Richard is building a rectangular backyard from 400 feet of fencing. The fencing must cover three sides of the backyard (the fourth side is bordered by Richard's house). What is the maximum area of this backyard?

Dec 8, 2020

#1
+1

The maximum will be a SQUARE (which is a rectangle )  of side length  400/3  ft

you should be able to calculate the area of the square now that you have the side length.....

Dec 8, 2020
#2
+345
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Yes, this is absolutely correct!

From here what we can do, since the area of a square is \$s^2\$, we can take \$400/3 = s\$ and just aquare it, and we get \$16\color{red}{00}00/9\$

Nacirema  Dec 8, 2020
edited by Nacirema  Dec 8, 2020
#3
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... 16000 / 9   sq ft

Guest Dec 8, 2020
#4
+1

....nope     160 000 / 9   ft 2

Guest Dec 8, 2020
#5
+345
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My bad, I had a typo in there. I corrected my answer. Thank you for catching it!

Nacirema  Dec 8, 2020
#6
+114090
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Let  two of the equal  sides  = x      Let the other side  = y

So

2x +  y  =  400   →     y = 400 - 2x

Area  = xy

A  =  x ( 400 - 2x)

A  = 400x  - 2x^2        take the  derivative

A' =  400 - 4x

Set  the  derivative  = 0

400  - 4x  =  0

400  = 4x

x = 100

y = 400  -2x  =   200

Max area =   100 * 200  =   20000 ft^2

See the graph  of the Area  function here.....https://www.desmos.com/calculator/pvjvhyciaz

Dec 8, 2020