Find the area of the largest rectangular garden that can be made so that one side of the
house serves as the natural boundary and 10 m of fencing material is required for the
remaining three sides.
A = x y
x + 2y = 10
so
x = 10 -2 y
A = 10 y - 2 y^2
dA/dy = 10 - 4 y
= 0 for min or max
y = 10/4 = 5/2
x = 10 - 5 = 5
A = 25/2
To find the area of the largest rectangular garden, we need to determine the dimensions of the rectangular garden.
Let's assume the length of the rectangular garden is L and the width is W.
Since one side of the house serves as the natural boundary, the length of the garden will be equal to the length of the house. Therefore, L = length of the house.
We know that 10 m of fencing material is required for the remaining three sides. In a rectangular garden, two sides will have the length L and one side will have the length W.
The perimeter of the rectangular garden can be calculated as:
Perimeter = 2L + W
Given that 10 m of fencing material is required for the remaining three sides, we can write the equation:
2L + W = 10
Solving this equation for W:
W = 10 - 2L
Now, we can find the area of the rectangular garden:
Area = Length × Width = L × W
Substituting the value of W, we get:
Area = L × (10 - 2L)
To find the maximum area, we need to determine the value of L that maximizes the area.
To do this, we can find the derivative of the area function and set it equal to zero:
d(Area)/dL = 10 - 4L = 0
Solving for L:
10 - 4L = 0
4L = 10
L = 10/4
L = 2.5
Now, we can calculate the width:
W = 10 - 2L
W = 10 - 2(2.5)
W = 10 - 5
W = 5
Therefore, the length of the rectangular garden is 2.5 m and the width is 5 m.
To calculate the area of the largest rectangular garden, we multiply the length and width:
Area = 2.5 × 5 = 12.5 square meters
So, the area of the largest rectangular garden that can be made is 12.5 square meters.
To find the area of the largest rectangular garden, we need to maximize the area. Let's go step by step.
1. Assume the length of the rectangular garden is x meters. Since one side of the house serves as a natural boundary, the width of the garden will be x meters.
2. The remaining three sides of the garden require 10 meters of fencing material. This means:
- Two sides of length x each
- One side of width x - 10 meters (because one side is taken by the house)
3. The total length of the fencing material is the sum of the lengths of these sides:
- Length: 2x
- Width: x - 10
4. Since the total length of the fencing material is given as 10 meters, we can write the equation:
2x + x - 10 = 10
5. Solve the equation for x:
3x - 10 = 10
3x = 20
x = 20/3
6. Now that we have the length (x), we can calculate the width:
Width = x - 10 = (20/3) - 10
Width = (20 - 30)/3
Width = -10/3
7. However, a negative width does not make sense in this context. We can ignore this value since it's not possible to have a rectangular garden with a negative width.
8. Therefore, the largest rectangular garden that can be made is when the length is 20/3 meters and the width is x - 10 meters.
9. Now, to find the area of the rectangular garden, multiply the length and width:
Area = (20/3) * (20/3 - 10)
10. Simplify the expression to get the area:
Area = (20/3) * (-10/3)
Area = -200/9
11. Similarly, negative area doesn't make sense. So, we need to ignore this value. The negative area can be ignored because it arises from the assumption that the garden has a width of x - 10 meters, which is not possible in this case.
Therefore, there is no feasible rectangular garden that can be made in this scenario.