The area of a rectangle is 100 square feet. If the perimeter is 58 feet, find the length and width of the rectangle.
Let's assume the length of the rectangle is L and the width is W.
According to the given information, the area of the rectangle is LW = 100.
Also, the perimeter of the rectangle is 2L + 2W = 58.
Now we have a system of equations:
LW = 100
2L + 2W = 58
To solve this system, we can use substitution or elimination. Let's use substitution method.
From the second equation, we can isolate one variable:
2L = 58 - 2W
L = 29 - W
Now we can substitute this expression for L in the first equation:
(29 - W)W = 100
29W - W^2 = 100
Rearranging the equation, we get:
W^2 - 29W + 100 = 0
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
We can factor the quadratic expression:
(W - 4)(W - 25) = 0
This equation has two solutions:
W - 4 = 0 or W - 25 = 0
If W - 4 = 0, then W = 4.
If W - 25 = 0, then W = 25.
So, the width of the rectangle can be either 4 feet or 25 feet.
Substituting these values back into the second equation, we can find the corresponding lengths:
If W = 4, then 2L + 2(4) = 58
2L + 8 = 58
2L = 50
L = 25
So, if W = 4, then L = 25.
If W = 25, then 2L + 2(25) = 58
2L + 50 = 58
2L = 8
L = 4
So, if W = 25, then L = 4.
Therefore, the length and width of the rectangle can be 25 ft by 4 ft or 4 ft by 25 ft.