You want to build a garden around your garden. One side borders your house so it does not need a fence. If you only have 50m of fencing what dimensions should you make your garden in order to maximize the area?
Let the side parallel to the house by y
the each of the other two sides be x
Make a sketch
2x + y = 50
y = 50-2x
area = xy
= x(50-2x)
=50x - 2x^2
the x of the vertex of this parabola is -b/(2a) = -50/-4 = 12.5
so x = 12.5
and y = 50-25 = 25
The garden should be 25 m long and 12.5 m wide
by Calculus:
area = 50x - 2x^2
d(area)/dx = 50-4x = 0 for a max of area
4x=50
x = 12.5
continue as above
To maximize the area of the garden with a limited amount of fencing, you would need to construct a rectangular garden shape. Since one side of the garden borders your house and does not require a fence, you will only need three sides to be fenced.
Let's analyze the problem step by step:
1. Determine the length of the side that borders the house:
Since one side of the garden already borders your house and doesn't require fencing, you can consider this length fixed. Let's say the length of this side is "x" meters.
2. Calculate the remaining fencing length:
The total fencing length you have is 50 meters. Since one side is already fixed, you need to find the remaining two equal sides to fence. So, the remaining fencing length would be 50 - x meters.
3. Formulate the area of the garden as a function:
The area of a rectangle can be calculated by multiplying the length and width. In this case, the length remains x (as it is the same as the side bordering the house), and the width is (50 - x)/2 since the remaining fencing should be divided equally between the other two sides.
Therefore, the area of the garden can be represented as: Area = x * (50 - x)/2.
4. Find the maximum area using calculus (optional):
To find the dimensions for the maximum area, you can differentiate the area function with respect to x, set it equal to zero, and solve for x. However, this step is optional for this problem since we can also solve it by considering the behavior of a quadratic equation.
5. Analyze the behavior of the quadratic equation:
The area equation expands to become: Area = (50x - x²)/2 = 25x - (x²/2).
This is a quadratic equation in the form of y = ax² + bx + c, where a = -0.5, b = 25, and c = 0.
The coefficient of x² (a) is negative, which indicates that the graph of the equation opens downwards. This means that the maximum area occurs at the vertex of the parabola.
6. Find the x-coordinate of the vertex:
The x-coordinate of the vertex can be found using the formula: x = -b / (2a). Substituting the values: x = -25 / (2 * -0.5) = -25 / (-1) = 25.
Since the length can't be negative, we discard the negative solution.
7. Calculate the corresponding width:
The width can be calculated using the expression: (50 - x)/2 = (50 - 25)/2 = 25/2 = 12.5.
Therefore, to maximize the area with 50 meters of fencing, you should make your rectangular garden dimensions 25 meters by 12.5 meters.