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.
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
You have a quadratic now:
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
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.
Now, remember your area formula?
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.
37694046292365
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
You have a quadratic now:
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
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)