The perimeter of a rectangle is 94 ft, and the area of the rectangle is 90ft ^2. What are the dimensions of the rectangle?
2p+2l = 94
p+l = 47
l = 47-p
pl = 90
p(47-p) = 90
-p^2 + 47p - 90 = 0
p^2 - 47p + 90 = 0
(p-45)(p-2) = 0
p = 45 or p = 2
if p=45 , l = 47-45 = 2
if p = 2, l = 47-2 = 45
the rectangle is 2 ft by 45 ft
To find the dimensions of the rectangle, we can set up a system of equations based on the given information.
Let's assume the length of the rectangle is L and the width of the rectangle is W.
We are given two pieces of information: the perimeter and the area of the rectangle.
1. Perimeter of a rectangle: The formula for the perimeter of a rectangle is given by 2L + 2W.
We are given that the perimeter of the rectangle is 94 ft.
So, we can write the equation as: 2L + 2W = 94.
2. Area of a rectangle: The formula for the area of a rectangle is given by L * W.
We are given that the area of the rectangle is 90 ft^2.
So, we can write the equation as: L * W = 90.
Now we have a system of two equations:
Equation 1: 2L + 2W = 94
Equation 2: L * W = 90
We can solve this system of equations to find the values of L and W.
One way to solve this system is by substitution:
Step 1: Solve Equation 1 for L:
2L + 2W = 94
2L = 94 - 2W
L = (94 - 2W) / 2
L = 47 - W
Step 2: Substitute the expression for L in Equation 2:
(47 - W) * W = 90
47W - W^2 = 90
Rearrange to quadratic form: W^2 - 47W + 90 = 0
Step 3: Solve the quadratic equation. You can use factoring, completing the square, or the quadratic formula. In this case, factoring the quadratic equation gives us:
(W - 5)(W - 42) = 0
Set each factor equal to zero:
W - 5 = 0 or W - 42 = 0
Solve for W:
W = 5 or W = 42
Step 4: Substitute the values of W back into Equation 1 to find the corresponding values of L:
If W = 5:
L = 47 - W = 47 - 5 = 42
If W = 42:
L = 47 - W = 47 - 42 = 5
Therefore, the dimensions of the rectangle can be either (L = 42 ft, W = 5 ft) or (L = 5 ft, W = 42 ft).