The length of a rectangle is 8 feet more than its width. If the width is increased by 4 feet and the length is decreased by 5 feet, the area will remain the same. Find the dimensions of the original rectangle.
L = w + 8
(L-5 )(w+4) = L w
(w+8-5)(w+4) = (w+8)w
(w+3)(w+4) = w^2 + 8 w
w^2 + 7 w + 12 = w^2 + 8 w
w = 12 etc
To find the dimensions of the original rectangle, let's break down the problem into steps:
Step 1: Set up the equations
Let's assume the width of the original rectangle is represented by 'w' (in feet).
According to the problem, the length of the rectangle is 8 feet more than its width, so we can represent the length as 'w + 8' (in feet).
The area of a rectangle is given by the formula: Area = length * width.
Step 2: Find the area of the original rectangle
We can use the given information to calculate the area of the original rectangle:
Area = (w + 8) * w
Step 3: Find the dimensions after the changes
The problem states that if the width is increased by 4 feet and the length is decreased by 5 feet, the area will remain the same. Let's calculate the new length and width:
New length = (w + 8) - 5 = w + 3
New width = w + 4
Step 4: Find the area of the new rectangle
The area of the new rectangle (after the changes) is:
New Area = (w + 3) * (w + 4)
Step 5: Set up the equation with the same area
According to the problem, the area remains the same. So we can set up the equation:
Area = New Area
(w + 8) * w = (w + 3) * (w + 4)
Step 6: Solve the equation
Now we can solve the equation to find the value of 'w' (width).
(w + 8) * w = (w + 3) * (w + 4)
w^2 + 8w = w^2 + 7w + 12
w^2 - w^2 + 8w - 7w = 12
w = 12
So, the width of the original rectangle is 12 feet.
Step 7: Find the length of the original rectangle
Using the width we found in Step 6, we can find the length:
Length = width + 8 = 12 + 8 = 20
Therefore, the dimensions of the original rectangle are 12 feet (width) and 20 feet (length).