Please show me step by step if possible:

Find dimensions of a rectanglar rug whose perimeter is 28 ft and whose area is 45 ft².

The length (longer side) equal ___ft and width (shorter side) equal ____ft.

P = 2L + 2W

Possible dimensions:

3 by 11
4 by 10
5 by 9
6 by 8
7 by 7

Which of those dimensions will give an area of 45 sq. ft?

To find the dimensions of a rectangular rug, we need to solve a system of equations using the given information.

Let's assume the length of the rug is 'l' feet and the width of the rug is 'w' feet.

Step 1: Write the equations based on the given information.

The perimeter of a rectangle is given by the formula: P = 2(l + w).
We are given that the perimeter is 28 ft, so we can write the equation as: 28 = 2(l + w).

The area of a rectangle is given by the formula: A = l * w.
We are given that the area is 45 ft², so we can write the equation as: 45 = l * w.

Step 2: Solve the system of equations.

We have two equations:
Equation 1: 28 = 2(l + w).
Equation 2: 45 = l * w.

Since Equation 1 contains 'l + w', we can solve it for 'l' or 'w' and substitute it into Equation 2.

From Equation 1, we get: l + w = 14.
If we solve it for 'l', we get: l = 14 - w.

Substituting this 'l' value into Equation 2, we have:
45 = (14 - w) * w.

Expanding and rearranging this equation, we get:
45 = 14w - w².

Step 3: Simplify and solve the quadratic equation.

Rearrange the equation:
w² - 14w + 45 = 0.

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use factoring in this example.

First, find two numbers whose product is 45 and whose sum is -14:
The numbers are -5 and -9 (because -5 * -9 = 45 and -5 + (-9) = -14).

Rewrite the equation using these numbers:
(w - 5)(w - 9) = 0.

Step 4: Determine the values of 'w' and 'l'.

From this factored equation, we have two possibilities:
w - 5 = 0 or w - 9 = 0.

If w - 5 = 0, then w = 5.
If w - 9 = 0, then w = 9.

Now, substitute these values back into Equation 1 to find the corresponding values of 'l'.

For w = 5:
l = 14 - w = 14 - 5 = 9.

For w = 9:
l = 14 - w = 14 - 9 = 5.

Therefore, the dimensions of the rectangular rug could be:
Length (longer side): 9 ft
Width (shorter side): 5 ft

Or

Length (longer side): 5 ft
Width (shorter side): 9 ft

4 by 9