How many different rectangular gardens can you make, whose dimensions are whole numbers, if you have 48 m. Of fencing materials and the width is at least 4 m? Consider a 6m by 18m rectangle the same as an 18m by 6m rectangle.
L + w = 24
20 + 4
19 + 5
18 + 6
17 + 7
16 + 8
15 + 9
14 + 10
13 + 11
12 + 12 seems to be nine
To determine the number of different rectangular gardens, we need to find all the possible combinations of length and width that satisfy the given conditions.
Let's start by setting up a system of equations based on the given information:
1. The perimeter of a rectangle is calculated by adding the lengths of all sides. Since the rectangle has 48 m of fencing material, the perimeter equation is: 2(length + width) = 48.
2. The width is stated to be at least 4 m, which indicates that the minimum value for the width is 4.
Now, let's solve the system of equations step by step:
1. Rearrange the perimeter equation to solve for length:
2(length + width) = 48
length + width = 24
length = 24 - width
2. Substitute the minimum value for the width (4) into the length equation:
length = 24 - 4
length = 20
This gives us one possible rectangle with dimensions 20m by 4m. However, we need to consider other possibilities as well.
3. Substitute the different values for the width into the length equation:
length = 24 - width
For width = 5:
length = 24 - 5
length = 19
For width = 6:
length = 24 - 6
length = 18
Continue this process until the width equals half the perimeter (24) since swapping the length and width does not create a different rectangle.
Now, let's calculate the number of unique rectangles:
We have found the following combinations:
- 20m by 4m
- 19m by 5m
- 18m by 6m
Since the problem states that a 6m by 18m rectangle is the same as an 18m by 6m rectangle, we only count one of them.
Therefore, there are three different rectangular gardens: 20m by 4m, 19m by 5m, and 18m by 6m.