Question

You make a rectangular quilt that is 5 feet by 4 feet. You use the remaining 10 square feet of fabric to add a border of uniform width to the quilt. What is the width of the border?

Short answer

The width of the border is 0.5 feet or 6 inches.

Step-by-step solution

Calculate initial area

Firstly, the initial area of the quilt is calculated. Since the quilt is rectangular, the formula will be \( Area = length*width \), so \( Area = 5feet * 4feet = 20 square feet \).

Calculate total area

Next, calculate the total area of the quilt with the added border. The total area will be the initial area plus the area of the remaining fabric, which is \( Total Area = 20 square feet + 10 square feet = 30 square feet \).

Find out the dimensions

Now, find out the new dimensions of the quilt with the border. Let the width of the border be x, the length of the quilt will be \( 5 feet + 2x \) and the width of the quilt will be \( 4 feet + 2x \). Hence the equation for the total area would be \( (5 feet + 2x) * (4 feet + 2x) = 30 square feet \).

Solve the quadratic equation

Then, solve the quadratic equation \( (5 feet + 2x) * (4 feet + 2x) = 30 square feet \) for the width of the border. Sorting out this equation will yield \( 20 + 5x + 8x + 4x^2 = 30 \). Therefore, treating it like a quadratic equation where \( 4x^2 + 13x - 10 = 0 \) which can be solved by factoring or using quadratic formula to get the positive root as \( x = 0.5 feet = 6 inches \) (negative value will be discarded as dimensions cannot be negative)

Key concepts

Algebraic Expressions

Understanding algebraic expressions is crucial when solving problems involving variables and constants. These expressions consist of numbers, variables (such as x and y), and the arithmetic operations of addition, subtraction, multiplication, and division.

For example, in the quilt exercise, we formulated an expression \(5 feet + 2x\)\ for the new length of the quilt, which included the initial length and twice the border width, since the border is added to both sides. Similarly, \(4 feet + 2x\)\ is the expression for the new width. These expressions represent the lengths of each side of the quilt after the border is added and are essential in understanding the next steps towards finding the width of the border.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are polynomial equations of the second degree, typically in the form \( ax^2 + bx + c = 0 \) where a, b, and c are constants. The formula is expressed as:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)\
This formula will give us two solutions since it includes the \( \pm \) sign, representing the possible positive and negative square roots. When solving the quilt problem, we ended up with the quadratic equation \( 4x^2 + 13x - 10 = 0 \)\, which would ideally be solved using this formula to find the value of x that represents the border width. In our case, we opted for factoring, as it was also possible. However, the quadratic formula could have been utilized if the equation proved hard to factor.

Factoring Quadratic Equations

Factoring is another technique for solving quadratic equations, usually simpler than the quadratic formula when the equation can easily be factored. It involves breaking down the equation into two binomials that, when multiplied, produce the original quadratic equation. In the context of our exercise, we transformed the equation for area calculation into a quadratic equation \(4x^2 + 13x - 10 = 0\)\ and factored it to find the possible values of x.

Factoring is preferable when the coefficients are small, and the equation is simple enough to spot potential factors quickly. If the equation were more complex, the quadratic formula or completing the square might be the more feasible routes to finding the solution.

Rectangular Area Calculation

The calculation of a rectangle's area is intrinsic to a variety of mathematical problems and is particularly fundamental when it comes to geometry. In the rectangular quilt exercise, we used the formula for the area of a rectangle, which is \( length \times width \)\.

At first, we calculated the initial area of the quilt, which was a simple application of this formula: \(5 feet \times 4 feet = 20 square feet\)\. We then had to consider the added border, which requires an understanding of how the border changes the dimensions of the rectangle to calculate the total area. Practical problems like these perfectly illustrate how algebra and geometry frequently intersect in real-world scenarios.