# Math

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A farmer has 230 ft of fence to enclose a rectangular garden. What is the largest garden area that can be enclosed with the 230 ft of fence? Explain your work.

• Math -

Hi Crystal,

Good question! I hope this explanation helps. :)

You know your farmer has 230 feet of fencing, so we're going to keep that.

You're looking for Area, so what is the area of a rectangle?

Area = x * y

But, first you need to define one of your variables in order to proceed. We do this by taking the perimeter of your rectangle. To get perimeter, you add the 4 sides, 2 of the length and 2 of the width.

Perimeter = 2x + 2y

Now, remember your 230 feet, that is the total perimeter you can possibly have because it's the maximum amount of fencing you have. Plug that in for P!

230 = 2x + 2y

Now, solve for one of your variables. Personally, I almost always solve for y because in a quadratic I prefer to work with x's.

So:

230 = 2x + 2y
-2x -2x

230 - 2x = 2y
_____________
2 (to get y alone)

115 - x = y

Great! Now you have defined one of your terms! You have a value for y. Plug that value for y in as y in your area formula, and solve.

A(x) = x * y
A(x) = x * (115 - x)
A(x) = 115x - x^2

A(x)= -x^2 + 115 x

Now, once you have it in this form, remember the form of a quadratic equation:

Ax^2 + Bx + C (A, B, and C are just your coefficients and they are integers)

To find your maximum area, you need to use this formula:

x = -B Here, B = 115
_____

2(A) Here, A = -1

So you have:

x = - 115
______
2 (-1)

x = -115
______
-2

• Math (Continued) -

x = -115
____
-2

Solve this to get: 57.5
So, your greatest value for x will be 57.5.

Plug this in to your perimeter equation to determine the value of y.

Remember:

P(x) = 230 = 2x + 2y
230 = 2(57.5) + 2y
230 = 115 + 2y
-115 -115
___________________
115 = 2y
___ ___
2 2

57.5 = y

You have a sqaure! You now know that the value of x that will produce your maximum area is 57.5 and your value for y that will produce maximum area is 57.5.

A(x) = x * y

Plug in your variables to find maximum area :)

A(x) = 57.5 * 57.5 = 3,306.25 Feet

Your maximum area = 3,306.25 Feet.

• Math -

37694046292365

• Math -

P=2x+2y(perimeter)
JUST TO ADD TO THE ABOVE SOLUTION.
230=2x+2y
2y=230-2x
y=115-x

A=xy(Area)
A=x[115-x]
A=115x-x^2
dA/dX=115-2X=0(at maximum)
115=2x
x=57.5(CRITICAL VALUE)

• Math -

105 feet by 105 feet

• Math -

57.5

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• Math -

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• Math -

A farmer wants to know the area of a garden that is 312312 meters by 125125 meters. He uses the given model for help. Which expression can be used to find the area in square meters?