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Chapter 0: Problem 36

A vacant rectangular lot is being turned into a community vegetable garden measuring 15 meters by 12 meters. A path of uniform width is to surround the garden. If the area of the lot is 378 square meters, find the width of the path surrounding the garden.

Short Answer

Expert verified
The width of the path surrounding the garden is 3 meters.

Step by step solution

01

Understand the problem and the given values

A rectangular garden has dimensions of 15m by 12m. A uniform path surrounds the garden and the total area inclusive of this path is 378sq m. It is required to find the width of the path.
02

Express width of the lot

The path extends on all sides of the rectangle. Let the width of the path be \(x\) meters. Hence, if we add the path's width to the length and width of the rectangle, it becomes \(15 + 2x\) and \(12 + 2x\) respectively. The factor 2 in \(2x\) is considered because the path extends around the garden.
03

Create a quadratic equation

We know, area = length * width. The area of the lot is \(378 = (15+2x) * (12+2x)\). This equation can be rewritten as a quadratic equation, \(4x^2 + 54x + 180 - 378 = 0\). Simplifying this equation results in \(4x^2 + 54x - 198 = 0\).
04

Solve the quadratic equation

The quadratic equation in the form \(ax^2 + bx + c = 0\) can be solved by the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the values of a, b and c in the equation we get \(x = \frac{-54 \pm \sqrt{54^2 - 4*4*(-198)}}{2*4}\) .Calculating the values, we get the solutions \(x = 3\) or \(x = -16.5\). Since width cannot be negative, we disregard the second solution.
05

Confirm the result

The width of the uniform path around the garden is 3 meters.

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