Word Problems: Border Problems 
Border problems generally require that you work with quadratic equations. Here is an example of this type of problem:
The Smiths’ have decided to put a paved walkway of uniform width around their swimming pool. The pool is a rectangular pool that measures 12 feet by 20 feet. The area of the walkway will be 68 square feet. Find the width of the walkway.
In order to solve quadratic equations involving maximums and minimums for rectangular regions, it is necessary to Let’s solve the example given in the introduction above.
The Smith’s have a rectangular pool that measure 12 feet by 20 feet. They are building a walkway around it of uniform width. First we need to draw a picture as illustrated in the figure below:
Next we need to write an equation.
Let x be the width of the walkway that will surround the pool. There are three rectangles in this picture on which we need to focus: the larger rectangle, the pool itself, and the rectangle that represents the walkway around the pool. Our equation will include the area of all three rectangles.
The width of the larger rectangle is , which simplifies to
width _{larger rectangle} =
area_{larger rectangle} =
By putting all three of these areas together, we know that the area of the larger rectangle is equal to the area of the pool plus the area of the walkway that surrounds the pool. So our equation becomes:
Finally, we need to solve this equation to find the width of the walkway.

Rearrange the terms for easier multiplication and find the sum of 68 and 240. 

Multiply the binomials. 

Combine like terms and subtract 308 from each side. 

Factor. 

Solve each factor. 

Since dimensions of a pool and a walkway around a pool cannot be negative our answer is that the width of the walkway is 1 foot. 
x = 1 
If regional building codes require that the walkway be 6 inches thick, how many square yards of cement must they purchase to complete the project? (concrete facts courtesy of Do It Yourself.com)
The volume of cement needed in cubic feet is (68 ft ^{2})(6/12 ft) = 34 ft ^{3}
^{
}
A typical 60pound bay of premixed concrete costs between $1.35$1.80 and yields onehalf of a cubic foot. They would need a minimum of 68 bags to complete the project: 68 x $1.80 = $122.40 plus a strong back.
One cubic yard equals (1 yd)^{3} = (3 ft)^{3} = 27 ft^{3}, so they will need 34 ft^{3}/27 ft^{3} = 1.26 yd^{3}.
A fullyloaded cement truck carries 10 cubic yards at an average price of $65 per cubic yard PLUS $17 per cubic yard short of the truck's full capacity. Since concrete is only sold in increments of full yards, no fractions, their cost would be $65 + $17(8) = $265.

Let's Practice








For problems of this type you must correctly draw and label a diagram, correctly write an equation to represent the given data, and correctly solve the equation for the solution to the problem. After solving the equation, you must substitute your solutions into the given data to reach the final answer.

D Saye


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