The length of a rectangular room is two feet less than three times its width. The area of the
rectangle is 176 square feet.
a) Write an algebraic equation to model this problem.
b) Solve the equation to find the dimensions of the rectangle.
Length = x
Width = y
a) x = 3y - 2
b) Area = Length*Width
= xy
=(3y-2)(y)
=> 176 = 3y^2 - 2y
Solve the quadratic for the dimensions.
How do I find x? I found y as y= 8, -22/3 but should I calculate x with the two solutions of y?
a) Let's assign the width of the rectangle as "w" feet. According to the problem, the length of the rectangle is two feet less than three times its width, which can be expressed as (3w - 2) feet.
The area of a rectangle can be calculated by multiplying its length and width. In this case, the area is given as 176 square feet.
So, the equation to model this problem can be written as:
w * (3w - 2) = 176
b) To solve the equation, we can start by simplifying the equation:
3w^2 - 2w - 176 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Since the equation can't be easily factored, we can use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 3, b = -2, and c = -176. Substituting these values into the quadratic formula:
w = (-(-2) ± √((-2)^2 - 4(3)(-176))) / (2(3))
= (2 ± √(4 + 2112)) / 6
= (2 ± √2116) / 6
= (2 ± 46) / 6
Now, we have two possible values for w:
w₁ = (2 + 46) / 6
= 48 / 6
= 8
w₂ = (2 - 46) / 6
= -44 / 6
= -7.33 (rounded to two decimal places)
Since width cannot be negative, we disregard w₂ as the width of the rectangle. Thus, the width of the rectangle is 8 feet.
Now, we can find the length of the rectangle using the given information that the length is two feet less than three times the width:
length = 3w - 2
= 3(8) - 2
= 24 - 2
= 22
Therefore, the dimensions of the rectangle are 8 feet by 22 feet.
a) To write an algebraic equation to model this problem, we can let "x" represent the width of the rectangular room.
According to the problem statement, the length of the room is two feet less than three times its width. So the length can be represented as "3x - 2".
The area of a rectangle is calculated by multiplying its length by its width. Since the area is given as 176 square feet, our equation can be formed as:
x*(3x - 2) = 176
b) To solve the equation and find the dimensions of the rectangle, we can follow these steps:
1. Distribute the x to get:
3x^2 - 2x = 176
2. Subtract 176 from both sides to set the equation equal to zero:
3x^2 - 2x - 176 = 0
3. Now, we can solve the quadratic equation. There are several methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. In this case, we'll use factoring.
This equation is not easily factored, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 3, b = -2, and c = -176.
Substituting these values into the quadratic formula, we get:
x = (-(-2) ± √((-2)^2 - 4(3)(-176))) / (2(3))
x = (2 ± √(4 + 2112)) / 6
x = (2 ± √(2116)) / 6
x = (2 ± 46) / 6
4. Simplify the expression:
x1 = (2 + 46) / 6 = 48 / 6 = 8
x2 = (2 - 46) / 6 = -44 / 6 = -22/3 (which is not a valid width)
Since the width of a room cannot be negative and should be a rational number, the width of the room is 8 feet.
Now, we can substitute the width (x) into the expression for the length:
Length = 3x - 2 = 3(8) - 2 = 24 - 2 = 22
Therefore, the dimensions of the rectangular room are:
Width = 8 feet
Length = 22 feet