Word Lesson: Border Problems

By D Saye

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

know how to solve quadratic equations
know formulas for finding the area of geometric figures
know how to multiply polynomials
write and solve an equation for the problem

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 length of the larger rectangle is , which simplifies to

length_{larger rectangle} =

The width of the larger rectangle is , which simplifies to

width_{larger rectangle} =

The area for the larger rectangle then becomes

area_{larger rectangle} =

The pool itself has an area of square feet.

The rectangle that wraps around the pool is given to be 68 square feet.
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 ft²)(6/12 ft) = 34 ft³
A typical 60-pound bay of pre-mixed concrete costs between $1.35-$ 1.80 and yields one-half 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 ft³, so they will need 34 ft³/27 ft³ = 1.26 yd³.

A fully-loaded 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.

A rectangular garden is surrounded by a walk of uniform width. The area of the garden is 180 square yards. If the dimensions of the garden plus the walk are 16 yards by 24 yards, find the dimensions of the garden.

A rectangular flower garden is surrounded by a walkway 4 meters wide. The flower garden is 6 meters longer than it is wide. If the total area is 576 square meters more than the area of the flower garden, find the dimensions of the flower garden.

The mat around a picture measures x inches across. The length of the picture with its mat is 40 inches, and the width of the picture with its mat is 24 inches. If the area of the picture is 192 square inches, find the dimensions of the picture.

24 inches by 8 inches
56 inches by 40 inches
48 inches by 16 inches
20 inches by 12 inches

A family plans to surround their pool with a patio of constant width. The pool has an area of 150 square feet. The dimensions of the pool with the patio will be 15 feet by 20 feet. Find the dimensions of the pool.

15 feet by 2.5 feet
10 feet by 7.5 feet
15 feet by 10 feet
10 feet by 5 feet

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.

	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