A rectangular dog pen is to be made to enclose an area of 625 sq. ft. The pen is to be divided into 2 equal sections by a section of the fence. Find the dimensions of the pen that will require the least amount of fencing material.
Thank you!
Calculus I assume?
Length*Width= 625
fencelenght= 2L + 3W
W= 625/L
fencelength= 2L+ 3*625/L
d fl/dl= 0=2 -3*625/L^2
or L= sqrt (3/2 * 625)
and W= 625/sqrt (3/2 625)
Now, if you are in algebra, I don't see any way of working this except graphing
fencelength=2L -325/L and looking for the minimum.
The local city council want to build a bike path around a square park. The park has an area of 9 square miles. How many miles long will the bike path be
To find the dimensions of the pen that will require the least amount of fencing material, we need to consider the shape of the pen. Let's assume that the length of the rectangle is L and the width of each section is W.
Since the pen is divided into two equal sections, each section would have an area of 625/2 = 312.5 sq. ft.
The total amount of fencing material required for the pen can be calculated by adding up the lengths of the four sides of the rectangle.
Perimeter of the pen = 2L + 4W
To minimize the amount of fencing material, we need to find the values of L and W that minimize this expression while still satisfying the given area condition.
We can solve this problem using calculus. First, let's express the perimeter as a function of a single variable, say L.
Perimeter(L) = 2L + 4(312.5 / L)
Now, to find the minimum, we need to find the value of L that makes the derivative of the perimeter function equal to zero. Let's differentiate the perimeter function with respect to L:
d(Perimeter) / dL = 2 - 1250 / L^2
Setting the derivative equal to zero:
2 - 1250 / L^2 = 0
Simplifying the equation:
2L^2 - 1250 = 0
Solving this quadratic equation gives us two possible values for L: L = ±√625. Since the length cannot be negative, we take the positive value:
L = √625 = 25 ft
Now that we have the value of L, we can solve for the width W by using the area formula:
W = (312.5) / L = (312.5) / 25 = 12.5 ft
Therefore, the dimensions of the pen that will require the least amount of fencing material are 25 ft by 12.5 ft.