Questions: The length of a rectangular floor is 5 feet less than twice its width. The area of the floor is 150 square feet. What is the length of the room? feet

Solution Steps

Let the width of the rectangular floor be \( w \) feet. According to the problem, the length of the floor is 5 feet less than twice its width. Therefore, the length can be expressed as:

\[ l = 2w - 5 \]

The area of the rectangle is given by the formula:

\[ \text{Area} = \text{length} \times \text{width} \]

Substituting the expressions for length and width, we have:

\[ 150 = (2w - 5) \times w \]

Expanding the equation:

\[ 150 = 2w^2 - 5w \]

Rearrange the equation to set it to zero:

\[ 2w^2 - 5w - 150 = 0 \]

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -5 \), and \( c = -150 \).

The quadratic formula is given by:

\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substitute the values of \( a \), \( b \), and \( c \):

\[ w = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \times 2 \times (-150)}}{2 \times 2} \]

\[ w = \frac{5 \pm \sqrt{25 + 1200}}{4} \]

\[ w = \frac{5 \pm \sqrt{1225}}{4} \]

\[ w = \frac{5 \pm 35}{4} \]

Calculate the two possible values for \( w \):

1. \( w = \frac{5 + 35}{4} = \frac{40}{4} = 10 \)
2. \( w = \frac{5 - 35}{4} = \frac{-30}{4} = -7.5 \)

Since width cannot be negative, we have \( w = 10 \).

Substitute \( w = 10 \) back into the expression for length:

\[ l = 2w - 5 = 2(10) - 5 = 20 - 5 = 15 \]

Final Answer