A rectangular closet is three feet longer than twice the width. The area of the closet is 65 square feet. Find the perimeter of the closet.
Width = X ft.
Length = 2x+3 Ft.
A = L*W = (2x+3)x = 65.
2x^2+3x = 65,
2x^2 + 3x - 65 = 0,
Use Quadratic Formula:
X = (-3 +- sqrt(9 + 520))/4
X = (-3 +- 23)/4 = 5, and -6.5.
X = 5 Ft. = Width.
Length = 2*5 + 3 = 13 Ft.
P = 2L + 2W.
To find the perimeter of the closet, we first need to determine the width and length of the closet.
Let's assume the width of the closet is represented by the variable 'w'.
According to the given information, the length of the closet is three feet longer than twice the width. So, we can write the length as 2w + 3.
The area of a rectangle is calculated by multiplying its length by its width. In this case, the area is given as 65 square feet. Therefore, we have the equation:
w * (2w + 3) = 65
Now we can solve this quadratic equation to find the value of 'w':
2w^2 + 3w - 65 = 0
We can solve this equation by factoring, completing the square, or using the quadratic formula. Factoring might not be easy in this case, so let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = 3, and c = -65. Plugging these values into the formula, we can calculate the width:
w = (-3 ± √(3^2 - 4 * 2 * -65)) / (2 * 2)
Simplifying the equation:
w = (-3 ± √(9 + 520)) / 4
w = (-3 ± √529) / 4
Since we are dealing with dimensions, the width cannot be negative. Therefore, we take the positive value:
w = (-3 + √529) / 4
w = (16 + 3) / 4
w = 19 / 4
w = 4.75 feet
Now that we have the width, we can calculate the length:
Length = 2w + 3
Length = 2 * 4.75 + 3
Length = 9.5 + 3
Length = 12.5 feet
Finally, we can find the perimeter by adding up all the sides of the rectangle:
Perimeter = 2 * (Length + Width)
Perimeter = 2 * (12.5 + 4.75)
Perimeter = 2 * 17.25
Perimeter = 34.5 feet
Therefore, the perimeter of the closet is 34.5 feet.