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.
Another tutor helped me and I was able to find the
y(width) solution set (y=8,-22/3) but I couldn't find the length
negative dimensions are extraneous solutions
... they have no "real world" meaning
divide the area by the width to find the length
To solve this problem, we can follow these steps:
a) Write an algebraic equation to model this problem.
Let's assume the width of the room is represented by "w" (in feet). According to the problem, the length of the room is two feet less than three times its width. So, the length can be represented by the expression "3w - 2" (in feet). The formula for calculating the area of a rectangle is length multiplied by width. Therefore, we can write the equation as:
Area = Length x Width
176 = (3w - 2)w
b) Solve the equation to find the dimensions of the rectangle.
To solve the equation, we need to simplify and rearrange it into the standard quadratic form. Let's start by expanding the equation:
176 = 3w^2 - 2w
Now, let's rearrange the equation and bring all the terms to one side:
3w^2 - 2w - 176 = 0
Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, since factoring may not be readily apparent, we can use the quadratic formula:
Given a quadratic equation of the form ax^2 + bx + c = 0, the quadratic formula states that the solution is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -2, and c = -176. Plugging these values into the quadratic formula, we get:
w = (-(-2) ± √((-2)^2 - 4*(3)*(-176))) / (2*(3))
w = (2 ± √(4 + 2112)) / 6
w = (2 ± √2116) / 6
w = (2 ± 46) / 6
Now we have two possible solutions for the width:
1) w = (2 + 46) / 6 = 48 / 6 = 8
2) w = (2 - 46) / 6 = -44 / 6 = -22/3
Let's check which solution is valid for the given problem statement. Since the width of a room cannot be negative, we can discard the negative solution. Therefore, the valid width of the room is 8 feet.
Now, let's find the length using the valid width of 8 feet. Substitute the width (w = 8) into the expression for the length:
Length = 3w - 2
Length = 3(8) - 2
Length = 24 - 2
Length = 22 feet
Therefore, the dimensions of the rectangle are width = 8 feet and length = 22 feet.