Questions: Solving a word problem using a quadratic equation with rational... The length of a rectangle is 5 m less than twice the width, and the area of the rectangle is 52 m^2. Find the dimensions of the rectangle. Length : m Width : m

Solution Steps

To solve this problem, we need to set up a quadratic equation based on the given information. Let the width of the rectangle be $w$. Then, the length of the rectangle can be expressed as $2w - 5$. The area of the rectangle is given by the product of its length and width, which equals 52 square meters. We can set up the equation $w(2w - 5) = 52$ and solve for $w$.

Let the width of the rectangle be $w$. The length of the rectangle is given as $2w - 5$. The area of the rectangle is given by the product of its length and width, which equals $52 \, \text{m}^2$. Therefore, we set up the equation: $$w(2w - 5) = 52$$

Rearrange the equation to standard quadratic form: $$2w^2 - 5w - 52 = 0$$ Solving this quadratic equation, we get the solutions: $$w = -4 \quad \text{and} \quad w = \frac{13}{2}$$ Since the width cannot be negative, we discard $w = -4$ and take: $$w = \frac{13}{2} = 6.5000$$

Using the width $w = 6.5000$, we calculate the length: $$\text{Length} = 2w - 5 = 2(6.5000) - 5 = 13.0000 - 5 = 8.0000$$

Final Answer