Colton is building a rectangular fence. He wants the length of the fence to be at least 25 yards and the perimeter to be no more than 150 yards. Write and solve a system of linear inequalities that shows all possible combinations of the dimension of the rectangle.
Let's denote the length of the fence as L and the width as W.
We know that the length must be at least 25 yards, so we can write the inequality:
L ≥ 25
The perimeter of a rectangle is given by:
Perimeter = 2(L + W)
We are also given that the perimeter must be no more than 150 yards:
2(L + W) ≤ 150
Combining these inequalities, we have the system of linear inequalities:
L ≥ 25
2(L + W) ≤ 150
Now we can solve for possible combinations of dimensions. Let's solve for W in terms of L in the second inequality:
2(L + W) ≤ 150
L + W ≤ 75
W ≤ 75 - L
So we have the following range of possible values for W in terms of L:
0 ≤ W ≤ 75 - L
Therefore, all possible combinations of dimensions for the rectangular fence that satisfy the given conditions are:
25 ≤ L
0 ≤ W ≤ 75 - L