The perimeter of a rectangle is 46 inches, and the area is 120 square inches. Find the length and width of the rectangle.
A = LW 120 = LW or 120/l = W
P = 2L + 2W
46 = 2L + 2(120/L)
multiply through by L
46L = 2L^2 + 240
2L^2 - 46L + 240 = 0
2(L^2 - 23L + 120) = 0
Factor and set each part equal to zero. Omit the negative value for x.
Once you have the Length substitute into the original equation to find Width.
To find the length and width of the rectangle, we can set up a system of equations using the given information.
Let's assume the length of the rectangle is "l" inches and the width is "w" inches.
We know that the perimeter of a rectangle is given by the formula: P = 2l + 2w.
Given that the perimeter of the rectangle is 46 inches, we can write the equation:
2l + 2w = 46
We also know that the area of a rectangle is given by the formula: A = l * w.
Given that the area of the rectangle is 120 square inches, we can write the equation:
l * w = 120
Now, we have a system of equations:
2l + 2w = 46 ----(Equation 1)
l * w = 120 ----(Equation 2)
There are different methods to solve this system of equations, such as substitution or elimination. Let's use the substitution method.
From Equation 1, we can solve for l in terms of w:
2l = 46 - 2w
l = (46 - 2w) / 2
l = 23 - w
Now, we substitute this expression for l in Equation 2:
(23 - w) * w = 120
23w - w^2 = 120
Rearranging this equation, we get a quadratic equation:
w^2 - 23w + 120 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula.
Factoring the equation, we have:
(w - 8)(w - 15) = 0
This means that w - 8 = 0 or w - 15 = 0.
So, we have two possible widths for the rectangle: w = 8 inches or w = 15 inches.
Substituting these values of w back into Equation 1, we can find the corresponding lengths:
When w = 8:
l = 23 - w
l = 23 - 8
l = 15
So, when w = 8, the length of the rectangle is 15 inches.
When w = 15:
l = 23 - w
l = 23 - 15
l = 8
So, when w = 15, the length of the rectangle is 8 inches.
Therefore, the length and width of the rectangle are either 15 inches by 8 inches or 8 inches by 15 inches.