the area of a rectangle is 425 square feet. if the perimeter is 84 feet, find the length and the width of the rectangle

To find the length and width of the rectangle when given the area and perimeter, we can use a system of equations.

Let's assume that the length of the rectangle is L and the width is W.

We know that the area of a rectangle is given by the formula: Area = Length * Width

In this case, the area of the rectangle is given as 425 square feet, so we can write the equation as:
L * W = 425 ----(Equation 1)

We also know that the perimeter of the rectangle is given by the formula: Perimeter = 2 * (Length + Width)

In this case, the perimeter is given as 84 feet, so we can write the equation as:
2 * (L + W) = 84 ----(Equation 2)

Now, we have a system of equations with two variables (L and W). We can solve this system to find the values of L and W that satisfy both equations.

To solve the system, we can use substitution or elimination method. Let's use substitution method:

From Equation 1, we can solve for L in terms of W:
L = 425 / W

Now, substitute this value of L into Equation 2:
2 * (425 / W + W) = 84

Simplify the equation:
2 * (425 + W^2) / W = 84

Multiply both sides of the equation by W to eliminate the denominator:
2 * (425 + W^2) = 84 * W

Expand and rearrange the equation:
850 + 2W^2 = 84W

Rearrange the equation in standard quadratic form:
2W^2 - 84W + 850 = 0

Now, we can solve this quadratic equation for W.

You can use the quadratic formula: W = (-b +/- sqrt(b^2 - 4ac)) / 2a

In this case, a = 2, b = -84, and c = 850.

Substitute these values in the quadratic formula to find the values of W. Once you have the value of W, substitute it back into Equation 1 to find the value of L.

Finally, you will have the length (L) and width (W) of the rectangle.

Eq1: L*W = 425Ft^2.

Eq2: 2L + 2W = 84Ft.

L + W = 42.
L e 42-W,

In Eq1, substitute 42-W for L:
(42-W)W = 425,
42W - W^2 - 425 = 0,
W^2 -42W + 425 = o,
(W-25)(W-17) = 0,

W-25 = 0,
W = 25.

W-17 = 0,
W = 17.

In Eq1 substitute 17 for W:
L*17 = 425,
L = 25 Ft., W = 17.

When L = 17, W = 25.