Answer:
29 by 70 meters
Step-by-step explanation:
You need to know how area and perimeter relate to the dimensions of the lawn.
The area formula is ...
A = LW . . . . . . . the product of length and width
The perimeter formula is ...
P = 2(L+W) . . . . twice the sum of length and width
The latter tells you the sum of length and width will be ...
P/2 = L + W = (198 m)/2 = 99 m
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It usually works well to assume that the dimensions are integers, then look for the factors of the area figure that give the right perimeter.
2030 = 2·5·7·29
= 2×1015 = 5×406 = 7×290 = 10×203 = 14×145 = 29×70 = 35×58
Of these factor pairs, the one that has a sum of 99 is (29, 70).
The dimensions of the lawn are 29 m by 70 m.
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You can also solve this graphically or algebraically. The graphical solution finds the points of intersection between LW=2030 and L+W=99, or the equivalent. Attached is a graph of the problem.
The algebraic solution can use substitution:
L = 99-W
2030 = (99 -W)(W)
W² -99W +2030 = 0 . . . . subtract the right side and put in standard form
W = (99±√(99² -4·2030))/2 . . . . . . use the quadratic formula
W = (99 +√1681)/2 = (99 ±41)/2 = {29, 70}
It doesn't matter which dimension you call the width.
The lawn is 29 m by 70 m.
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The quadratic formula tells you the solution to ...
[tex]ax^2+bx+c=0[/tex]
is given by ...
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Here, we have a=1, b=-99, c=2030.
Of course (-99)² = 99², and 4(1)(2030) = 4·2030, so we have simplified the solution a little bit when writing it above.
