Problem 146 - College Algebra

Question

A rectangular parking lot has a length that is 3 yards greater than the width. The area of the parking lot is 180 square yards. Find the length and the width.

Step-by-Step Solution

Answer

The width of the parking lot is 12 yards and the length is 15 yards.

Step 1: Define the variables

Define the width of the parking lot as \(x\), then the length will be \(x + 3\). Here, \(x\) represents the width and \(x + 3\) represents the length.

Step 2: Formulate the equation

The area of a rectangle is given by its width times its length. So, in this case the equation will be \(x(x + 3) = 180\).

Step 3: Solve the equation for x

Solving the equation \(x(x + 3) = 180\) results in a quadratic equation. This equation can be rewritten as \(x^2 + 3x - 180 = 0\). Solving this quadratic equation for its roots will give the value for \(x\).

Step 4: Solve the quadratic equation

Applying quadratic formula, \(x = [-b ± sqrt(b^2 - 4ac)] / 2a\), to solve \(x^2 + 3x - 180 = 0\) where here, \(a = 1, b = 3, c = -180\), we get \(x = [-3 ± sqrt((3)^2 - 4*1*(-180))] / 2*1\). Solving this, we arrive at two possible values for \(x\), 12 and -15. Since a width can't be negative, we discard -15, so the width, \(x\), is 12 yards.

Step 5: Find the length of the parking lot

Now, substituting \(x = 12\) back into the length expression will give us the length of the parking lot. So, we find the length to be \(12 + 3 = 15\) yards.