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.