Problem 21 - Algebra 2 and Trigonometry

Question

The ratio of the length to the width of a rectangle is 7 : 3. The area of the rectangle is 336 square centimeters. What are the dimensions of the rectangle?

Step-by-Step Solution

Answer

The dimensions of the rectangle are 28 cm by 12 cm.

Step 1: Define Variables

Let the length of the rectangle be \(7x\) and the width be \(3x\), because the ratio of length to width is 7:3.

Step 2: Write the Area Formula

The formula for the area of a rectangle is \( ext{Area} = ext{length} \times ext{width} \). Substitute the expressions for length and width: \(7x \times 3x = 336\).

Step 3: Simplify the Equation

Simplify the equation from Step 2: \(21x^2 = 336\).

Step 4: Solve for x

Divide both sides of the equation by 21 to solve for \(x^2\): \(x^2 = \frac{336}{21}\). This simplifies to \(x^2 = 16\).

Step 5: Find the Value of x

Take the square root of both sides: \(x = 4\) (since \(x\) must be positive).

Step 6: Calculate Dimensions

Substitute \(x = 4\) back into the expressions for length and width: length = \(7 \times 4 = 28\) cm and width = \(3 \times 4 = 12\) cm.

Key Concepts

Rectangle Dimensions

Understanding rectangle dimensions starts with the fundamental properties of a rectangle, which has two pairs of parallel sides. These sides include the length and the width. In many problems, like the given exercise, dimensions are expressed as a ratio. Here, the ratio is 7:3 indicating that for every 7 units of length, there are 3 units of width. Thus, you can represent the length as \(7x\) and the width as \(3x\). This variable 'x' is a scaling factor that maintains the ratio but adapts to the actual size measurements required by the problem's conditions. The ratio remains consistent, but 'x' determines the real-world dimensions of your rectangle. Recognizing this relationship is key to solving dimension-related questions effectively.

Area Calculation

The area of a rectangle gives us a measure of how much space is enclosed within its sides. It's calculated using the formula:

• \( \text{Area} = \text{length} \times \text{width} \)

For the exercise here, substituting \(7x\) for length and \(3x\) for width into the formula leads us to the equation \(7x \times 3x = 336\). This acts as a bridge between our knowns (the area) and unknowns (the exact dimensions). Solving this equation requires simplifying it to \(21x^2 = 336\). Once simplified, you can proceed to isolate \(x\) by applying algebraic techniques, enabling you to find the specific dimensions.

Algebraic Expressions

Algebraic expressions make it possible to represent real-world problems in a structured format that can be solved systematically. In the exercise provided, expressions like \(7x\) and \(3x\) encapsulate an entire relationship within a simple format. These expressions empower us to manipulate equations, allowing for simplification and eventual resolution. By rewriting the area formula as \(21x^2 = 336\), you've utilized algebra to facilitate problem-solving. Solving for \(x\) involves dividing both sides by \(21\) and subsequently taking the square root, yielding \(x = 4\). Once 'x' is determined, substituting back into expressions \(7x\) and \(3x\) gives precise dimensions. This process of manipulation showcases the power and versatility of algebra in solving ratio-based geometric problems.