The perimeter of this rectangular portion needs to be 14 yards and the diagonal is 5 yards. Can you help Mike determine the length and width of this new portion? Mike requested that you show him how you arrive at the answers.
The perimeter is the distance around the rectangle. Let's try a width of 3 yards and a length of 4 yards.
Then use the Pythagorean theorem to see if the diagonal is 5 yards.
To find the length and width of the rectangular portion, we can use a system of equations considering the perimeter and the diagonal.
Let's assume the length of the new portion is L and the width is W.
1. Firstly, we know that the perimeter of a rectangle is the sum of all its sides. In this case, the perimeter is given as 14 yards.
The formula for the perimeter is:
Perimeter = 2L + 2W
Substituting the given values, we get:
14 = 2L + 2W
2. Secondly, we can use the Pythagorean theorem to relate the diagonal, length, and width of a rectangle.
The Pythagorean theorem is:
diagonal^2 = length^2 + width^2
Substituting the given values, we have:
5^2 = L^2 + W^2
25 = L^2 + W^2
Now, we have a system of equations:
Equation 1: 14 = 2L + 2W
Equation 2: 25 = L^2 + W^2
To solve this system, we can use one of various methods. Let's solve it using substitution.
From equation 1, we can isolate one of the variables, let's say W, in terms of L:
14 = 2L + 2W
2W = 14 - 2L
W = 7 - L
Now we substitute this expression for W in equation 2:
25 = L^2 + (7 - L)^2
25 = L^2 + 49 - 14L + L^2
25 = 2L^2 - 14L + 49
2L^2 - 14L + 24 = 0
Let's factorize it:
2(L^2 - 7L + 12) = 0
2(L - 3)(L - 4) = 0
Now, we have two possible solutions for L: L = 3 or L = 4.
To find the corresponding width, we substitute the values of L back into our original equation (Equation 1) and solve for W:
For L = 3:
14 = 2(3) + 2W
14 = 6 + 2W
2W = 14 - 6
2W = 8
W = 4
So one possible solution is L = 3 and W = 4.
For L = 4:
14 = 2(4) + 2W
14 = 8 + 2W
2W = 14 - 8
2W = 6
W = 3
So another possible solution is L = 4 and W = 3.
Therefore, Mike can have a rectangular portion with dimensions either 3 yards by 4 yards or 4 yards by 3 yards.