Problem 18 - Elementary Algebra

Question

The length of a rectangle is four ninths its width. The area is 144 square feet. Find the dimensions.

Step-by-Step Solution

Answer

Answer: The dimensions of the rectangle are Length = 8 feet and Width = 18 feet.

Step 1: Write the equations based on the given information

We are given that the length of the rectangle is four ninths times its width. We can write this as: Length = (4/9) * Width We are also given that the area of the rectangle is 144 square feet. The area of a rectangle is calculated as Length * Width, so we can write this as: Area = Length * Width = 144

Step 2: Substitute the expression for Length into the Area equation

In the previous step, we found the equation for Length in terms of Width: Length = (4/9) * Width Now let's substitute this expression for Length into the Area equation: Area = ((4/9) * Width) * Width = 144

Step 3: Solve for Width

Now we have an equation with only one variable, Width, which we can solve: (4/9) * Width^2 = 144 To solve for Width, first multiply both sides of the equation by 9: 4 * Width^2 = 1296 Next, divide both sides of the equation by 4: Width^2 = 324 Now, take the square root of both sides: Width = 18 feet

Step 4: Find the Length

We now have the value for Width (18 feet). We can find the Length by substituting the value of Width into the equation for Length: Length = (4/9) * Width = (4/9) * 18 Length = 8 feet Both sides of the rectangle have a nonnegative length, and the dimensions of the rectangle are Length = 8 feet and Width = 18 feet.

Key Concepts

Rectangle Dimensions

Understanding the dimensions of a rectangle is crucial when working with geometric figures and solving area problems. A rectangle is a four-sided figure with opposite sides that are equal in length and every angle is a right angle, meaning it is 90 degrees. The dimensions of a rectangle are referred to as the length (the longer side) and the width (the shorter side).

To calculate the area of a rectangle, you use the formula:
\[ \text{Area} = \text{Length} \times \text{Width} \]
The area is expressed in square units because it represents the amount of two-dimensional space inside the rectangle. When a problem includes a relationship between the length and width—like one dimension being a fraction of the other—it sets the stage for creating algebraic equations to solve for the unknown dimensions.

Algebraic Equations

Algebraic equations are mathematical statements that use variables, numbers, and operation signs to express a relationship between quantities. They are solved by finding the value(s) of the variables that make the equation true. For instance, in our problem, we have two quantities related to each other: the length and width of a rectangle.

By translating words into an algebraic expression, like 'the length is four ninths its width', we create an equation that looks like: \[ \text{Length} = \left(\frac{4}{9}\right) \times \text{Width} \]
Algebraic equations are tools that help us describe and understand relationships in a precise, numerical way, which is essential in many areas of mathematics and science.

Quadratic Equations

Quadratic equations are a type of polynomial equation of the second degree, meaning they include a variable raised to the power of two. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants and \( x \) is the unknown variable. In area problems involving rectangles, quadratic equations often arise when equating the area to the product of length and width, especially when one dimension is a fraction or multiple of the other. After substituting and simplifying, you are left with a quadratic equation in terms of a single variable.

Solving a quadratic equation might involve rearranging terms, factoring, or applying the quadratic formula. In the context of our problem, we end up with an equation like \( (4/9) \times \text{Width}^2 = 144 \), which simplifies down to a quadratic equation set to zero.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \). For example, since \( 18 \times 18 = 324 \), we say that the square root of 324 is 18, written as \( \sqrt{324} = 18 \).

When solving quadratic equations like the one we have in our rectangle problem, taking the square root of both sides of the equation is often the penultimate step (after isolating the squared term). However, it is important to remember that there are usually two solutions to a square root, the positive and negative values. In the context of geometry, we typically use the positive root because lengths cannot be negative. This process robustly ties together our understanding of the algebra involved in geometrical shapes and their properties.