If I have a rectangular yard with an area of 48 square meters. And I put a fence post every 2 meters around the perimeter of the yard. But I only have 14 fence post.
What would be the dimensions for the yard?
Let's assume the length of the yard is L meters and the width is W meters.
The area of a rectangle is given by A = L * W. From the given information, we have A = 48 square meters.
The perimeter of a rectangle is given by P = 2L + 2W. We know that fence posts are placed every 2 meters along the perimeter, so the total number of fence posts required is equal to P/2 = L + W.
Given that you only have 14 fence posts, we can write the equation: L + W = 14.
Now, let's solve this system of equations.
1. From A = L * W, substitute A = 48:
48 = L * W --> equation (1)
2. From L + W = 14, solve for L:
L = 14 - W --> equation (2)
Substitute equation (2) into equation (1):
48 = (14 - W) * W
48 = 14W - W^2
0 = W^2 - 14W + 48
0 = (W - 6)(W - 8)
Now, we have two possible values for W: W = 6 or W = 8.
If W = 6, substitute this value into equation (2):
L = 14 - 6
L = 8
So, if the width is 6 meters and the length is 8 meters, the dimensions of the yard would be 8 meters by 6 meters, with an area of 48 square meters.
If W = 8, substitute this value into equation (2):
L = 14 - 8
L = 6
So, if the width is 8 meters and the length is 6 meters, the dimensions of the yard would still be 8 meters by 6 meters, with an area of 48 square meters.
Therefore, the dimensions of the yard would be 8 meters by 6 meters.
To determine the dimensions of the yard, we need to calculate the perimeter. Knowing that the fence posts are placed every 2 meters along the perimeter of the yard, and there are a total of 14 fence posts, we can calculate the perimeter.
Let's assume the length of the yard is L meters and the width is W meters.
The number of fence posts on the longer side of the rectangle is L/2, and the number of fence posts on the shorter side is (W-2)/2. The total number of fence posts is obtained by adding these two values and setting it equal to 14:
L/2 + (W-2)/2 = 14
Multiplying through by 2 to eliminate the fractions:
L + W - 2 = 28
Rearranging the equation:
L + W = 30
We also know that the area of the yard is 48 square meters, so we have:
L * W = 48
We now have a system of two equations:
L + W = 30
L * W = 48
By solving this system of equations, we can find the dimensions of the yard.
Let's substitute L = 30 - W into the second equation:
(30 - W) * W = 48
Expanding and simplifying:
30W - W^2 = 48
Rearranging:
W^2 - 30W + 48 = 0
Factoring, we get:
(W - 24)(W - 2) = 0
So, W can be either 24 or 2.
If W = 24, then L = 30 - 24 = 6.
If W = 2, then L = 30 - 2 = 28.
Therefore, there are two possible dimensions for the yard:
1. 6 meters by 24 meters.
2. 28 meters by 2 meters.
To determine the dimensions of the yard, we need to find the length and width that result in an area of 48 square meters.
Let's assume the length of the yard is L meters and the width is W meters. Then, the area can be calculated by multiplying the length and width: L * W = 48.
Now, let's find the possible dimensions of the rectangular yard by factoring the number 48.
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
We will start trying different combinations of L and W, using the factors we found, to see if any of them result in a perimeter that can be covered by the available 14 fence posts.
Let's determine the possible combinations and their perimeters:
1) L = 1 meter, W = 48 meters
Perimeter = 2L + 2W = 2(1) + 2(48) = 2 + 96 = 98 meters
2) L = 2 meters, W = 24 meters
Perimeter = 2L + 2W = 2(2) + 2(24) = 4 + 48 = 52 meters
3) L = 3 meters, W = 16 meters
Perimeter = 2L + 2W = 2(3) + 2(16) = 6 + 32 = 38 meters
4) L = 4 meters, W = 12 meters
Perimeter = 2L + 2W = 2(4) + 2(12) = 8 + 24 = 32 meters
5) L = 6 meters, W = 8 meters
Perimeter = 2L + 2W = 2(6) + 2(8) = 12 + 16 = 28 meters
Based on the options above, only the combination with L = 6 meters and W = 8 meters has a perimeter of 28 meters, which can be covered by the available 14 fence posts.
Therefore, the dimensions of the yard would be 6 meters by 8 meters.