An enclosed rectangular garden is 6 meters by 15 meters. Additional fencing within the garden can divide it into three smaller plots, two of them square and one rectangular. What is the least number of meters of additional fencing that is needed?
Make a sketch,
so the two squares must be
6by6 each and the rectangle must be 3by6
Wouldn't we just need 2 divisions of 6 m each?
or 12 m ?
Am I missing something here ?
To find the least number of meters of additional fencing needed, we need to determine the arrangement that minimizes the additional fencing required.
Let's break down the problem step by step:
1. The initial garden is rectangular with dimensions 6 meters by 15 meters.
2. We want to divide the garden into three smaller plots, two squares, and one rectangular.
3. Let's assume the side length of each square plot is "x" meters.
4. Therefore, the rectangular plot will have dimensions (15 - 2x) meters by (6 - x) meters.
5. The total amount of additional fencing required will be the perimeter of each plot minus the perimeter of the initial garden.
Now, let's calculate the perimeters step by step:
- The perimeter of the initial garden is 2 * (6 + 15) = 42 meters.
- The perimeter of each square plot is 4 * x meters.
- The perimeter of the rectangular plot is 2 * ((15 - 2x) + (6 - x)) = 42 - 6x meters.
Finally, we can calculate the additional fencing needed:
- Additional fencing needed = (perimeter of square plot 1) + (perimeter of square plot 2) + (perimeter of rectangular plot) - (perimeter of initial garden)
= 4x + 4x + (42 - 6x) - 42
= 8x - 6x
= 2x meters.
Therefore, the least number of meters of additional fencing needed is 2x meters.
To find the least number of meters of additional fencing needed, we need to determine the dimensions of the three smaller plots and calculate the perimeter of each plot.
Let's start by visualizing the original rectangular garden and the three smaller plots within it.
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┌─────────┬───────┐
│ │ │
│ │ │
│ │ │
│ │ │
│ │ │
└─────────┴───────┘
```
We know that the original garden is 6 meters by 15 meters. Now, let's divide the garden into three smaller plots.
```
┌───────┬───────┐
│ │ │
│ │ │
│ A │ B │
│ │ │
│ │ │
├───────┼───────┤
│ │ │
│ C │ │
│ │ │
└───────┴───────┘
```
Plot A is a square, plot B is a square, and plot C is a rectangle. Let's assume the sides of the square plots are x meters each, and the length and width of the rectangular plot C are y meters and z meters, respectively.
Now, we can set up equations based on the given information:
1. The dimensions of the original garden:
- Length: 15 meters
- Width: 6 meters
2. The dimensions of the three smaller plots:
- Plot A (square): x meters by x meters
- Plot B (square): x meters by x meters
- Plot C (rectangle): y meters by z meters
Using the given information, we can derive additional equations:
1. The total length of the garden is divided into the lengths of the three plots:
- 15 = x + x + y
2. The total width of the garden is equal to the width of plot C:
- 6 = z
To find the least number of meters of additional fencing needed, we need to determine the values of x, y, and z.
Let's solve the equations one by one:
1. From the equation 15 = x + x + y, we can simplify it to 15 = 2x + y.
- By setting x = 0, we can find the maximum value for y: 15 = 2(0) + y => y = 15.
Therefore, the maximum value for y is 15.
2. From the equation 6 = z, we can see that z is already given as 6.
Now that we have determined the values of x, y, and z, we can calculate the perimeters of each plot to find the least amount of additional fencing needed.
Perimeter of plot A (square): P(A) = 4x meters
Perimeter of plot B (square): P(B) = 4x meters
Perimeter of plot C (rectangle): P(C) = 2y + 2z meters
Substituting the known values, we get:
P(A) = 4x = 4(0) = 0 meters (since x = 0)
P(B) = 4x = 4(0) = 0 meters (since x = 0)
P(C) = 2y + 2z = 2(15) + 2(6) = 30 + 12 = 42 meters
To find the minimum additional fencing needed, we add up the perimeters of the plots:
Total additional fencing needed = P(A) + P(B) + P(C) = 0 + 0 + 42 = 42 meters
Therefore, the least number of meters of additional fencing that is needed is 42 meters.