Questions: A rectangle has a length of 17 inches less than 7 times its width. If the area of the rectangle is 2204 square inches, find the length of the rectangle.

Solution Steps

Given that the length \( l \) of the rectangle is 17 inches less than 7 times its width \( w \), we can express this relationship as: \[ l = 7w - 17 \] The area \( A \) of the rectangle is given as 2204 square inches, which leads to the equation: \[ A = l \cdot w = 2204 \] Substituting the expression for \( l \) into the area equation gives: \[ (7w - 17) \cdot w = 2204 \]

Expanding the equation results in: \[ 7w^2 - 17w - 2204 = 0 \] This is a standard quadratic equation in the form \( ax^2 + bx + c = 0 \) where \( a = 7 \), \( b = -17 \), and \( c = -2204 \).

To find the values of \( w \), we calculate the discriminant: \[ D = b^2 - 4ac = (-17)^2 - 4 \cdot 7 \cdot (-2204) = 62001 \] Since the discriminant is positive, we can find two possible solutions for \( w \): \[ w = \frac{-b \pm \sqrt{D}}{2a} = \frac{17 \pm \sqrt{62001}}{14} \] Calculating the two possible values for \( w \): \[ w_1 = \frac{17 + \sqrt{62001}}{14} \quad \text{and} \quad w_2 = \frac{17 - \sqrt{62001}}{14} \] Since \( w_2 \) will yield a negative value, we discard it and take \( w_1 \).

Using the positive width \( w_1 \), we can find the length \( l \): \[ l = 7w - 17 \] Substituting \( w_1 \) into this equation gives us the length of the rectangle.

Final Answer