Questions: Select the correct answer. A civil engineer is designing a public parking lot for the new town hall. The number of cars in each row should be six less than the number of rows in the lot. The mayor has requested that the new lot should hold twice as many cars as the current parking lot, which has space for 56 cars. Determine which equation the civil engineer can use to find the number of rows, x, he should include in the new parking lot. A. x^2 - 6x = 112 B. 2x^2 - 12x = 112 C. x^2 - 6 = 112 D. 6x^2 - 56 = 112

Solution Steps

To solve this problem, we need to set up an equation based on the given conditions. Let \( x \) be the number of rows. According to the problem, the number of cars in each row is \( x - 6 \). The total number of cars in the new parking lot is then \( x \times (x - 6) \). This total should be twice the number of cars in the current parking lot, which is 56. Therefore, the equation becomes \( x \times (x - 6) = 2 \times 56 \). Simplifying this equation will help us identify the correct option.

The problem states that the number of cars in each row is six less than the number of rows, \( x \). Therefore, the number of cars per row is \( x - 6 \). The total number of cars in the parking lot is given by the product of the number of rows and the number of cars per row, which is \( x(x - 6) \).

The new parking lot should hold twice as many cars as the current parking lot, which has space for 56 cars. Therefore, the total number of cars in the new parking lot should be \( 2 \times 56 = 112 \).

We equate the expression for the total number of cars in the new parking lot to 112: \[ x(x - 6) = 112 \]

Expanding the equation, we have: \[ x^2 - 6x = 112 \] Rearranging gives: \[ x^2 - 6x - 112 = 0 \] Solving this quadratic equation, we find the solutions: \[ x = -8 \quad \text{or} \quad x = 14 \]

Since the number of rows, \( x \), must be a positive integer, we discard the negative solution. Thus, the number of rows is \( x = 14 \).

Final Answer