A rectangular room has an area of 400 sq. ft. The length is 7 ft less than twice the width. Find the dimensions of the room.
w(2w-7) = 400
To solve this problem, we can represent the width of the room as 'x' feet.
According to the given information, the length of the room is 7 feet less than twice the width.
So, the length would be 2x - 7 feet.
The area of a rectangle is calculated by multiplying its length and width.
Given that the area of the room is 400 square feet, we have the equation:
Area = Length × Width
400 sq. ft = (2x - 7) ft × x ft
Now, let's solve the equation step by step:
400 = (2x - 7) × x
400 = 2x² - 7x
To simplify the equation, let's move all the terms to one side and form a quadratic equation:
2x² - 7x - 400 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
Factoring the quadratic equation, we get:
(2x + 25)(x - 16) = 0
Setting each factor to zero and solving for 'x', we have:
2x + 25 = 0 or x - 16 = 0
For the first case, 2x + 25 = 0, if we solve for 'x', we get:
2x = -25
x = -12.5
Since we're measuring the width of a room, we can't have a negative value.
Therefore, we can conclude that x - 16 = 0:
x = 16
So, the width of the room is 16 feet.
To find the length, we can use the equation we derived earlier:
Length = 2x - 7
Plugging in the value of 'x' (16) into the equation, we find:
Length = 2(16) - 7
Length = 32 - 7
Length = 25
Therefore, the dimensions of the room are:
Width = 16 feet
Length = 25 feet