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Chapter 1: Problem 146

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.

Short Answer

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

Step by step solution

01

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.
02

Formulate the equation

The area of a rectangle is given by its width times its length. So, in this case the equation will be \(x(x + 3) = 180\).
03

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 \(x\).
04

Solve the quadratic equation

Applying quadratic formula, \(x = [-b 卤 sqrt(b^2 - 4ac)] / 2a\), to solve \(x^2 + 3x - 180 = 0\) where here, \(a = 1, b = 3, c = -180\), we get \(x = [-3 卤 sqrt((3)^2 - 4*1*(-180))] / 2*1\). 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.
05

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.

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