Problem 71 - Elementary Algebra

Question

The diagonal of a television monitor measures 32 inches. If the monitor has a 3: 2 aspect ratio, then determine its length and width. Round off to the nearest hundredth.

Step-by-Step Solution

Answer

Width: 26.61 inches, Height: 17.74 inches.

Step 1: Understand the Aspect Ratio

The aspect ratio is 3:2, meaning that the width to height ratio is 3:2. Let's denote the width as \( 3x \) and the height as \( 2x \).

Step 2: Set Up the Diagonal Equation

Using the Pythagorean theorem on the monitor's dimensions, the diagonal is given by \( \sqrt{(3x)^2 + (2x)^2} = 32 \).

Step 3: Simplify the Equation

Simplify the equation: \[ \sqrt{9x^2 + 4x^2} = 32 \] which becomes \[ \sqrt{13x^2} = 32 \].

Step 4: Solve for \( x \)

Square both sides to solve for \( x \): \[ 13x^2 = 1024 \].\Divide by 13: \[ x^2 = \frac{1024}{13} \].\So \[ x = \sqrt{\frac{1024}{13}} \].

Step 5: Calculate \( x \)

Compute \( x \): \[ x \approx \sqrt{78.769} \approx 8.87 \].

Step 6: Determine Length and Width

Use \( x \) to find the width and height.\Width \( = 3x \approx 3 \times 8.87 = 26.61 \).\Height \( = 2x \approx 2 \times 8.87 = 17.74 \).

Step 7: Round to the Nearest Hundredth

The length (width) is approximately 26.61 inches and the width (height) is approximately 17.74 inches.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics. It relates the lengths of the sides of a right triangle. According to the theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is often expressed as:

• \( a^2 + b^2 = c^2 \)

Where \( a \) and \( b \) are the lengths of the triangle's legs, and \( c \) is the length of the hypotenuse.
In the context of the problem, the television monitor is considered as a right triangle, with its diagonal as the hypotenuse. Using the Pythagorean Theorem, we can calculate the relationship between the diagonal and the width and height of the monitor. Understanding this theorem is crucial as it allows us to set up the equation needed to solve for the dimensions.

Algebraic Equations

Algebraic equations are mathematical statements that use letters or variables to represent numbers. In solving for unknowns in geometry problems, algebraic equations are instrumental.
To solve for the width and height of the television monitor, we use the equation derived from the Pythagorean Theorem:

• \( \sqrt{(3x)^2 + (2x)^2} = 32 \)

Simplifying this results in an equation:

• \( \sqrt{13x^2} = 32 \)

To find \( x \), we square both sides. This gives us a simpler algebraic equation,
\( 13x^2 = 1024 \),which can be manipulated to solve for \( x \). Understanding how to rearrange and simplify equations is crucial in tackling such problems.

Problem Solving

Problem solving in mathematics involves several steps that help break down complex problems into manageable parts. First, we need to comprehensively understand the problem. Identifying the relationship between different pieces of information, like the aspect ratio and the equation setup in this exercise, is essential.
In this case, the aspect ratio of 3:2 means we can express the monitor's width as \( 3x \) and height as \( 2x \). With the diagonal given, using the Pythagorean Theorem enables us to set up an equation.
Next, simplify the information using algebraic methods. Once the equation is simplified to \( 13x^2 = 1024 \), we find \( x \) and utilize it to compute the actual width and height. Rounding off to the nearest hundredth ensures precise answers, useful for practical applications like measuring television sizes.
Breaking the problem into these actionable steps allows solutions to emerge systematically.