Aldo has feet of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing. What is the maximum area that the garden can have?

missing info

how many feet of fencing?

Suppose he has 120 feet, change the solution below to the appropriate number of feet in your question

let length of field be y, (y being parallel to house.)
let the width be x

2x + y = 120
y = 120-2x

area = xy
= x(120-2x)
= 120x - 2x^2

If you are in Calculus .....
d(area)/dx = 120 - 4x = 0 for a max area
x = 30
y = 120-2(30) = 60
Max area = 30(60) = 1800 ft^2

if you don't take Calculus, complete the square
area = -2(x^2 - 60x + 900 - 900 )
= -2(x-30)^2 + 1800

max area is 1800, when x = 30

Considering all rectangles with a given perimeter, one side being another straight boundry, the 3 sided

rectangle enclosing the greatest area has a length to width ratio of 2:1.

To find the maximum area of the garden, we need to determine the dimensions of the rectangular garden that will use up all the fencing.

Let's assume that the length of the garden is x feet. Since we need to form three sides of the rectangle, two sides will each have a length of x feet, and the third side will have a length equal to the remaining fencing after deducting the lengths of the two sides with length x.

Since each side of length x will contribute to the fencing twice, the equation representing the fencing can be written as:

2x + y = Aldo's total fencing

As we are trying to find the maximum area, we need to maximize the length and width of the garden. Since the length already has a maximum value of x, we can substitute y with Aldo's total fencing minus 2x.

Substituting this value into the equation, we get:

2x + (Aldo's total fencing - 2x) = Aldo's total fencing

Simplifying this equation, we can solve for x:

2x - 2x + Aldo's total fencing = Aldo's total fencing

0x = 0

This equation tells us that x can be any value (as long as it's positive) and the area will still be the maximum.

Therefore, the maximum area of the garden can be obtained when the length and width are equal, resulting in a square-shaped garden.

In conclusion, the maximum area of the garden will be achieved when all sides are equal in length, and the area will be x * x (since it's a square).

To find the maximum area of the garden, we need to determine the dimensions of the rectangular garden that will use up the entire length of fencing and still have the largest area possible.

Let's assume the length of the garden is L, and the width is W. Since the fencing will form three sides of the garden, we can calculate the amount of fencing used using the perimeter formula:

Perimeter = 2L + W

Since we know that Aldo has a certain number of feet of fencing, we can set up an equation using the given information:

2L + W = feet of fencing

Now, we can solve for W:

W = feet of fencing - 2L

To find the maximum area, we need to maximize the value of L * W. Since W is dependent on L, we can substitute the equation for W into the area formula:

Area = L * (feet of fencing - 2L)

Now, we have an equation that expresses the area of the garden as a function of L. To find the maximum area, we need to find the critical points of this function's graph.

To do this, we take the derivative of the area function with respect to L and set it equal to zero:

d/dL (Area) = 0

Next, we solve this equation to find the critical points.

Once we find the values of L that satisfy the equation, we substitute them back into the area function to find the maximum area.

Let me know if you need help understanding any part of this process.