Question 1089216: A man has 30 feet of fence for a rectangular garden. What is the area of the largest garden he can build with whole-number dimensions?
Found 2 solutions by Theo, rothauserc:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 30 feet is the perimeter.
the perimeter of the garden is rectangular, so let x = the length and y equal the width and let p equal the perimeter.
you get p = 2x + 2y
the area is equal to x * y.
set a equal to area and you get:
a = x * y
in the equation for perimeter, solve for y to get:
y = (30 - 2x) / 2
simplify to get y = 15 - x
in the formula for area, replace y with 15-x to get:
a = x * y = x * (15 - x)
simplify to get:
a = 15x - x^2
this is a quadratic equation that can be solved for the maximum or minimum value.
since the coefficient of the x^2 term is negative, we will be solving for the maximum value.
convert the equation to standard form to get:
-x^2 + 15x = area.
the equation is now in standard form.
when in standard form, a is the coefficient of the x^2 term and b is the coefficient of the x term and c is the constant term.
a is equal to -1
b is equal to 15
c = 0 (if it's not there, it's value is equal to 0)
the maximum value of the area will be when x = -b/2a
therefore the maximum value of the area is at x = -15/-2 = 7.5
x, however, needs to be an integer, therefore the maximum value will be when x is equal to 7 or when x is equal to 8.
since the quadratic equation is symmetrical about the max/min value, the value of the area should be the same.
when x = 7, the equation of 2x + 2y = 30 becomes 14 + 2y = 30
solve for y to get y = 8.
when x = 8, the equation of 2x + 2y = 30 becomes 16 + 2y = 30
solve for y to get y = 7.
therefore, the maximum area is when x = 7 or 8 and the corresponding value of y = 8 or 7.
when x or y = 7 and 8, the area is 56.
that's because x * y = area becomes 7 * 8 = 56 or becomes 8 * 7 = 56
you can construct a table to show you that this is true.
start with 2x + 2y = 30
divide both sides of this equation by 2 to get:
x + y = 15
since x and y both have to be integers, the only possible values are:
x x = 1 and y = 14
x = 2 and y = 13
x = 3 and y = 12
x = 4 and y = 11
x = 5 and y = 10
x = 6 and y = 9
x = 7 and y = 8
and the reverse from x = 8 to x = 15
example:
x = 8 and y = 7
x = 9 and y = 6
x = 10 and y = 5
etc.
when x = 1 and y = 14, the area is 14
when x = 2 and y = 13, the area is 26
when x = 3 and y = 12, the area is 36
when x = 4 and y = 11, the area is 44
when x = 5 and y = 10, the area is 50
when x = 6 and y = 9, the area is 54
when x = 7 and y = 8, the area is 56 ***** maximum area
when x = 8 and y = 7, the area is 56 ***** maximum area
when x = 9 and y = 6, the area is 54
when x = 10 and y = 5, the area is 50
etc.
bottom line is you can do it by formula or you can do it by grunt work.
either way you get maximum area is 56.
Answer by rothauserc(4718)  (Show Source):
You can put this solution on YOUR website! let l be length and w be width
:
1) Area(A) of rectangle = l * w
:
2) 2l + 2w = 30 feet(perimeter of rectangular garden)
:
solve equation 2 for l
:
l + w = 15
l = 15 - w
:
now substitute for l in expression for Area
:
(15-w) * w
15w -w^2
:
this is the equation of a parabola that curves downward, therefore we could take the first derivative to find the coordinates of the vertex
:
the expression in standard form is
:
-w^2 +15w
:
the first derivative is -2w + 15
:
-2w +15 = 0
-2w = -15
w = 7.5 feet
:
we could use the equation for x coordinate(w in this case) of the vertex
w = -b / 2a = -15 / (-2 * 1) = 7.5
:
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the max area is 8 * 7 = 56 feet
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