Sam has 1200 feet of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. Express the area of the field as a function of its dimension. Find the dimensions of the field that has the largest area.
Well, Sam certainly has quite the fencing dilemma! But fear not, I'm here to help you solve this problem with a smile on your face.
Let's call the width of the field "w" and the length of the field "l". Since the field borders a straight river, we only need to fence off three sides. This means that the total amount of fencing used will be the perimeter of the field, which is given as 1200 feet.
Now, the perimeter of a rectangle is the sum of all its sides, which in this case would be:
2w + l = 1200
To make things a little simpler, let's express l in terms of w by subtracting 2w from both sides of the equation:
l = 1200 - 2w
The area of a rectangle is simply the product of its length and width, so the area of the field (A) can be expressed as:
A = l * w
Substituting the value of l from the previous equation:
A = (1200 - 2w) * w
Now, all we have to do is find the dimensions of the field that will give us the largest possible area. To do this, we'll take the derivative of the area function with respect to w and set it equal to zero, to find the maximum point.
dA/dw = 1200 - 4w
Setting this derivative equal to zero:
1200 - 4w = 0
Simplifying, we find:
4w = 1200
w = 1200/4
w = 300 feet
Now, we can substitute this value of w back into the equation for l that we found earlier:
l = 1200 - 2w
l = 1200 - 2(300)
l = 1200 - 600
l = 600 feet
So, the dimensions of the field that will give us the largest possible area are 300 feet for the width and 600 feet for the length.
Let's denote the width of the rectangular field as x and the length as y.
Since there is no need for a fence along the river, the total length of the fence required is the sum of the width and twice the length: 2x+2y.
We know that the total fencing available is 1200 feet, so we can write the equation:
2x + 2y = 1200.
To find the area of the field, we multiply the width (x) by the length (y): A = x * y.
Now, let's express the length (y) in terms of the width (x) using the equation 2x + 2y = 1200:
2y = 1200 - 2x
y = 600 - x.
Substituting this value for y in the equation for the area:
A = x * (600 - x).
To find the dimensions of the field that maximize the area, we need to find the value of x that maximizes the function A = x * (600 - x).
To find the maximum value, we can take the derivative of the function with respect to x and set it equal to zero:
dA/dx = 0,
d(x(600 - x))/dx = 0.
Simplifying this equation:
600 - 2x = 0,
2x = 600,
x = 300.
Substituting this value back into the equation for the length (y):
y = 600 - x,
y = 600 - 300,
y = 300.
So, the dimensions of the field that maximize the area are x = 300 feet and y = 300 feet.
To find the maximum area, we substitute these values back into the equation for the area:
A = x * y,
A = 300 * 300,
A = 90000 square feet.
Therefore, the maximum area of the field is 90000 square feet.
To express the area of the field as a function of its dimensions, let's denote the length of the field as "L" and the width of the field as "W".
Since the field borders a straight river, we do not need to fence along that side. Therefore, we only need to fence three sides of the field: two sides with lengths L and one side with length W.
The total length of the fencing required will be:
2L (for the length sides) + W (for the width side) = 2L + W
We know that Sam has 1200 feet of fencing, so we can therefore write the equation:
2L + W = 1200
To find the dimensions of the field that maximize the area, we need to express the area A as a function of the dimensions L and W.
The formula for the area of a rectangle is:
A = Length x Width
So, A = L x W
Now, since we have an equation for W in terms of L from above (W = 1200 - 2L), we can substitute this expression for W into the area formula:
A = L x (1200 - 2L)
Expanding this expression, we get:
A = 1200L - 2L^2
To find the dimensions of the field that maximize the area, we need to find the value of L that maximizes the function A = 1200L - 2L^2. This can be done by finding the vertex of the parabola defined by this quadratic function.
To find the vertex, we can use the formula:
L = -b / 2a
In this case, a = -2 and b = 1200, so:
L = -1200 / (2 * -2) = -1200 / -4 = 300
Therefore, the length of the field that maximizes the area is L = 300 feet.
To find the corresponding width, we can substitute this value of L into the equation we derived for W:
W = 1200 - 2L = 1200 - 2 * 300 = 1200 - 600 = 600
Therefore, the width of the field that maximizes the area is W = 600 feet.
To summarize, the dimensions of the field that have the largest area are:
Length = 300 feet
Width = 600 feet
And the corresponding maximum area A = Length x Width = 300 x 600 = 180,000 square feet.
length parallel to river --- x ft
each of other two sides -- y ft
2x + y = 1200 ----> y = 1200 - 2x
area = xy
= x(1200-2x)
= 1200x -2 x^2
since you labeled it "Calculus"
d(area)/dx = 1200 - 4x
= 0 for a max of area
4x = 1200
x = 300
width = 300
length = 1200 - 2(300) = 600