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.
L = 45/W
2L + 2W = 28
Substitute 45/W for L in second equation and solve for W. Insert that value into the first equation and solve for L. Check by inserting both values into the second equation.
To find the dimensions of a rectangular rug, follow these steps:
Step 1: Understand the problem.
We are given the perimeter and the area of the rectangular rug. We need to find the length (longer side) and width (shorter side) of the rug.
Step 2: Set up the equations.
Let's denote the length as L and the width as W.
We know that the perimeter (P) of a rectangle is given by the equation:
P = 2L + 2W
We also know that the area (A) of a rectangle is given by the equation:
A = L * W
Step 3: Write the given information as equations.
We are given that the perimeter is 28 ft, so we can rewrite the perimeter equation as:
28 = 2L + 2W
We are also given that the area is 45 ft², so we can rewrite the area equation as:
45 = L * W
Step 4: Solve the system of equations.
We have two equations and two unknowns. We can use substitution or elimination to solve this system of equations. Let's use substitution.
Let's solve the perimeter equation for L in terms of W:
2L = 28 - 2W
L = (28 - 2W) / 2
L = 14 - W
Substitute this value of L into the area equation:
45 = (14 - W) * W
45 = 14W - W²
Step 5: Rearrange the area equation.
To solve for W, let's rearrange the equation:
W² - 14W + 45 = 0
Step 6: Solve the quadratic equation.
Now we need to find the values of W that satisfy this equation.
We can either factor the quadratic equation or use the quadratic formula.
Factoring the equation:
(W - 5)(W - 9) = 0
Setting each factor equal to zero:
W - 5 = 0 or W - 9 = 0
W = 5 or W = 9
So, we have two potential values for W: W = 5 or W = 9.
Step 7: Find the corresponding values of L.
Now, substitute these values of W back into the equation L = 14 - W to find the corresponding values of L.
For W = 5:
L = 14 - 5
L = 9
For W = 9:
L = 14 - 9
L = 5
Step 8: Determine the dimensions of the rectangular rug.
The dimensions of the rectangular rug are:
Length (L) = 9 ft and Width (W) = 5 ft (or vice versa).
So, the length (longer side) is 9 ft and the width (shorter side) is 5 ft.