The length of a rectangle is 4 inches more than its width. The area of the rectangle is equal to 5 inches more than 2 times the perimeter. Find the length and width of the rectangle.
area is in^2, perimeter is inches. So the question is very poorly worded. Still,
w(w+4) = 5 + 2*2(w + w+4)
solve for w and then w+4.
To solve this problem, we need to set up equations based on the given information and then solve them.
Let's start by assigning variables to the unknowns. Let's say the width of the rectangle is 'w' inches. According to the given information, the length of the rectangle is 4 inches more than its width, so we can say the length is 'w + 4' inches.
Now, let's set up equations using the information we have. The area of a rectangle is given by the formula A = length * width. In this case, the area is equal to 5 inches more than 2 times the perimeter, which can be expressed as 2P + 5, where P represents the perimeter.
So, the equation for the area of the rectangle becomes:
A = (w + 4) * w = 2P + 5
We also know that the perimeter of a rectangle is given by the formula P = 2(length + width). In this case, the perimeter can be expressed as P = 2(w + (w + 4)), which simplifies to P = 4w + 8.
Now we have two equations: A = (w + 4) * w = 2P + 5 and P = 4w + 8.
We can now substitute the expression for P into the equation for A:
(w + 4) * w = 2(4w + 8) + 5
Expanding the equation:
w^2 + 4w = 8w + 16 + 5
Simplifying:
w^2 + 4w = 8w + 21
Rearranging the equation to have zero on one side:
w^2 - 4w - 8w - 21 = 0
Combining like terms:
w^2 - 12w - 21 = 0
Now we can solve this quadratic equation. We can factor the equation or use the quadratic formula to find the values of 'w'. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = -12, and c = -21.
Plugging in the values:
w = (-(-12) ± √((-12)^2 - 4 * 1 * -21)) / (2 * 1)
Simplifying and calculating:
w = (12 ± √(144 + 84)) / 2
w = (12 ± √(228)) / 2
w = (12 ± √(4 * 57)) / 2
w = (12 ± 2√(57)) / 2
w = 6 ± √(57)
So, the possible values of 'w' are 6 + √(57) and 6 - √(57).
Now, since the width of the rectangle cannot be negative (as it represents a physical length), we discard the negative solution. Therefore, 'w' = 6 + √(57).
Using this value, we can calculate the length:
Length = w + 4
Length = (6 + √(57)) + 4
Length = 10 + √(57) inches
Therefore, the width of the rectangle is approximately 10 + √(57) inches, and the length is approximately 6 + √(57) inches.