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How Do You Find The Dimensions When You Know The Perimeter And The Square Feet?

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Robbi Boone Profile
Robbi Boone answered
Charles planted a garden using 36 feet of fencing for the perimeter.  The area of Charles's garden is 45 square feet. What are the dimensions of his garden?
Oddman Profile
Oddman answered
Let P and A stand for the perimeter and area, respectively. Let L and W stand for the length and width, respectively.
We know ...
P = 2(L+W)
A = LW

The first equation can be solved for L, which can then be substituted into the second equation.
P/2 = L + W (divide both sides by 2)
P/2 - W = L (subtract W from both sides)
A = (P/2 - W)W (substitute for L)
A = WP/2 - W2 (use the distributive property)
W2 - WP/2 + A = 0 (add W^2 - WP/2 to both sides to put the quadratic into standard form)
W = (-(-P/2) ±√((-P/2)2 - 4(1)(A)))/(2(1))
= ((P/2) ±√((P2-16A)/4)/2
W = (P±√(P2-16A))/4
These two solutions turn out to be L and W.

Example
P=36 ft, A=45 ft2
W = (36±√(362-16*45))/4
= (36±√576)/4
= (36±24)/4
= {3, 15}
The width is 3 ft, the length is 15 ft.

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