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?
Perimeter = 230 Ft. The max. area will
occur as we approach a square (57.5)^2=
3306.25 sq. ft.). Since we are required to use a rectangle, the largest area we can get using whole
numbers is: A = LW = 58(57) = 3306 sq, ft. P = 58(2) + 57(2) = 230 ft as
required.
If a farmer has 280 feet of fence and wants to build a rectangle that the width is two-thirds of the length how do you calculate the deminsions?
2. You are installing a new pre-constructed fence in front of your house. Each fence section measures 4 1/2 feet wide, and each end will also have a decorative piece that measures 1 3/4 feet wide.
If the space for the fence is 30 feet wide, what is the most number of 4 1/2 foot fence sections you could install?
I calculated 4 1/2 + 3/4=5.25
30/5.25= 4
What are the largest and smallest areas that can be made with 100 yards of fencing
To find the largest garden area that can be enclosed with the given amount of fence, we need to use the concept of optimization.
Let's assume the length of the rectangular garden is x feet. In that case, the width of the garden would be (230 - 2x) feet since the perimeter of a rectangle is given by the formula P = 2L + 2W, where L and W represent the length and width, respectively.
The formula for the area of a rectangle is A = L * W. Substituting the values for L and W in terms of x, we have:
A = x * (230 - 2x)
To find the largest possible area, we need to find the maximum value of this function.
Taking the derivative of the area function with respect to x and setting it equal to zero, we can find the critical points:
dA/dx = 230 - 4x = 0
Solving this equation for x, we get:
4x = 230
x = 230/4
x = 57.5
The critical value for the length is x = 57.5 feet.
To determine if this is a maximum or minimum, we can take the second derivative of the area function:
d^2A/dx^2 = -4
Since the second derivative is negative, this confirms that the critical point x = 57.5 is indeed the maximum.
Now we can substitute this value back into the area function to find the maximum area:
A = 57.5 * (230 - 2 * 57.5)
A = 57.5 * (230 - 115)
A = 57.5 * 115
A = 6612.5
Therefore, the largest garden area that can be enclosed with 230 ft of fence is 6612.5 square feet.