You have 32 feet of fencing and want to determine the maximum area that can be enclosed with the fencing to make a flower garden. You do not want to deal with fractions or decimals, so you are limiting the dimensions to whole numbers. Also, the area is to be rectangular (or square) in shape.
not much of a problem here ...
the largest rectangle is obtained when that rectangle is a square, so
4 equal sides equal 32 feet,
each side is 8 feet
area is 64 square ft
Considering all rectangles with a given perimeter, which one encloses the largest area?
The traditional calculus approach would be as follows.
Letting P equal the given perimeter and "x" the short side of the rectangle, we can write for the area A = x(P - 2x)/2 = Px/2 - x^2.
Taking the first derivitive and setting equal to zero, dA/dx = P/2 - 2x = 0, x becomes P/4.
With x = P/4, all four sides are equal making the rectangle a square.
.....The short side is P/4.
.....The long side is (P - 2(P/4))/2 = P/4.
Therefore, it can be unequivicaly stated that of all possible rectangles with a given perimeter, the square encloses the maximum area.
To determine the maximum area that can be enclosed with the given 32 feet of fencing, we need to find the dimensions of a rectangular (or square) garden that requires the least amount of fencing.
Let's assume the length of the rectangular garden is L and the width is W.
The perimeter of a rectangle is given by the formula: perimeter = 2L + 2W.
Given that the perimeter is 32 feet, we can write the equation as: 2L + 2W = 32.
Now, since we want to determine the maximum area, we need to express the area in terms of one variable. The area of a rectangle is given by the formula: area = length x width.
We can express the area as: area = L x W.
To make our job easier, let's rearrange the perimeter equation to express one variable in terms of the other. Let's isolate L: 2L = 32 - 2W.
Dividing both sides of the equation by 2: L = 16 - W.
Now, we can substitute this value of L into the area equation:
area = (16 - W) x W.
Expanding the equation: area = 16W - W^2.
To determine the maximum area, we can take the derivative of the area equation with respect to W, set it equal to zero, and solve for W.
Let's find the derivative of the area equation: d(area)/dW = 16 - 2W.
Setting the derivative equal to zero: 16 - 2W = 0.
Solving for W: 2W = 16, W = 16/2, W = 8.
Therefore, the width of the rectangle that will result in the maximum area with the given 32 feet of fencing is 8 feet.
Substituting this value of W back into the perimeter equation: 2L + 2(8) = 32.
Simplifying the equation: 2L + 16 = 32.
Subtracting 16 from both sides: 2L = 16, L = 16/2, L = 8.
Therefore, the length of the rectangle that will result in the maximum area with the given 32 feet of fencing is also 8 feet.
Hence, to enclose the maximum area with 32 feet of fencing, the dimensions of the rectangular flower garden should be 8 feet by 8 feet. The maximum area will be 8 feet x 8 feet = 64 square feet.
To determine the maximum area that can be enclosed with the given 32 feet of fencing, we need to consider the properties of a rectangular or square shape.
Let's start by assuming the shape is a rectangle with two sides of length "x" feet and two sides of length "y" feet. According to the problem, the perimeter of this rectangle should be 32 feet, so we can write the equation:
2x + 2y = 32
Simplifying the equation, we have:
x + y = 16
We want to find the maximum area, which is given by the formula A = x * y. To avoid fractions or decimals, we need to search for whole numbers that satisfy the equation x + y = 16.
One way to do this is by trying different values for one variable, and then calculating the corresponding value for the other variable. Let's start with assuming x = 1 and calculate y:
1 + y = 16 (subtracting x from both sides)
y = 16 - 1
y = 15
So, if one side is 1 foot long, the other side would be 15 feet long. The corresponding area would be A = 1 * 15 = 15 square feet. Let's try some other values:
If x = 2, then y = 16 - 2 = 14, and A = 2 * 14 = 28 square feet.
If x = 3, then y = 16 - 3 = 13, and A = 3 * 13 = 39 square feet.
If x = 4, then y = 16 - 4 = 12, and A = 4 * 12 = 48 square feet.
We continue this process, trying different values for x and calculating the corresponding value for y, until we find the maximum area. By observing the values, we can notice that the area keeps increasing until x = 8. At x = 9, we will have y = 16 - 9 = 7, resulting in a smaller area than when x = 8.
Thus, the maximum area that can be enclosed with 32 feet of fencing is achieved when the shape is a rectangle with sides measuring 8 feet and 8 feet. The area would be A = 8 * 8 = 64 square feet.