a rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides. if the total area is 384 square feet, find the dimensions of the study area that will minimize the total length of the fence. how much fencing will be required?
xy=384
so, y = 384/x
f = 2x+3y = 2x + 1152/x
df/dx = 2 - 1152/x^2
min f is when df/dx=0, at x=24
So, the field is 24x16
f = 48+48=96
as usual in these problems, the fencing is divided equally among lengths and widths.
Ah, the study area and the fence, a tale as old as time. Let's put on our mathematical clown noses and solve this!
To minimize the total length of the fence, we want to find the dimensions of the study area that create a square. Why? Because a square with equal sides will always have the smallest possible perimeter for a given area.
Since the total area is 384 square feet, let's find the square root of that: √384 ≈ 19.60.
So the length and width of the study area that will minimize the total length of the fence are both approximately 19.60 feet.
Now let's calculate the fencing required. Since we need to divide the square into two equal parts, we only need to fence three sides: the top, the bottom, and the line along the division.
The length of the top and bottom sides is 19.60 feet each, and the length of the line along the division is the width of the square, which is also 19.60 feet.
Total fencing needed = 2 * (19.60 + 19.60) + 19.60 = 78.40 + 19.60 = 98 feet.
So, you'll need approximately 98 feet of fencing. Have fun with your study area, and remember, sometimes laughter is the best study companion!
To minimize the total length of the fence, the study area should be a square shape. Let's call the length and width of the study area as x.
Given that the total area is 384 square feet, we can set up the following equation:
x * x = 384
Now, let's solve for x by taking the square root of both sides:
x = √384
Using a calculator, we find:
x ≈ 19.60 feet (rounded to 2 decimal places)
Since the study area is a square shape, both the length and the width will be x.
Thus, the dimensions of the study area that will minimize the total length of the fence are approximately 19.60 feet by 19.60 feet.
To calculate the total fence length required, we need to find the perimeter of the study area (which is also the total length of the fence):
Perimeter = 2 * (length + width)
Substituting the values we found:
Perimeter = 2 * (19.60 + 19.60)
Perimeter = 2 * 39.20
Perimeter = 78.40 feet
Therefore, approximately 78.40 feet of fencing will be required to enclose the study area.
To find the dimensions of the study area that will minimize the total length of the fence, we need to use calculus to find the critical points of the function representing the total length of the fence.
Let's denote the length of the rectangle as L and the width as W. Since the fence runs along the division parallel to one of the sides, the width of each half of the rectangle will be W/2.
The total area of the rectangle is given as 384 square feet: L * W = 384.
We need to minimize the total length of the fence, which consists of the sum of the lengths of all sides of the rectangle. The fence will have two sides of length L and two sides of length W/2.
The total length of the fence can be calculated as: F = L + L + W/2 + W/2 = 2L + W.
To proceed, we need to express L in terms of W using the area equation.
Rearrange the area equation: L = 384 / W.
Substitute this expression for L into the equation for the total length of the fence: F = 2(384 / W) + W.
The equation for the total length of the fence now only contains a single variable W, allowing us to find its minimum value.
To find the minimum, take the derivative of the equation with respect to W and set it equal to zero:
dF/dW = -2 * 384 / W^2 + 1 = 0.
Solve the equation for W:
-2 * 384 / W^2 + 1 = 0,
-2 * 384 = W^2,
W^2 = 2 * 384,
W^2 = 768,
W = sqrt(768).
Now that we have the width, we can substitute it back into the equation for L to find the corresponding length:
L = 384 / sqrt(768).
The dimensions of the study area that will minimize the total length of the fence are approximately: L ≈ 16.97 feet and W ≈ 27.71 feet.
To find the amount of fencing required, we need to calculate the total length of the fence. Using the values we found above:
F = 2L + W = 2 * 16.97 + 27.71 ≈ 61.65 feet.
Therefore, approximately 61.65 feet of fencing will be required.