Problem 80 - Algebra and Trigonometry

Question

TV Monitors Two television monitors sitting beside each other on a shelf in an appliance store have the same screen height. One has a conventional screen, which is 5 in. wider than it is high. The other has a wider, high-definition screen, which is 1.8 times as wide as it is high. The diagonal measure of the wider screen is 14 in. more than the diagonal measure of the smaller. What is the height of the screens, correct to the nearest 0.1 in?

Step-by-Step Solution

Answer

The height of the screens is approximately 29.6 in.

Step 1: Define Variables

Let the height of both screens be \( h \). Let the width of the conventional screen be \( w_1 = h + 5 \), and the width of the wider, high-definition screen be \( w_2 = 1.8h \).

Step 2: Calculate Diagonal of Conventional Screen

The diagonal of a rectangle can be found using the Pythagorean theorem: \( d_1 = \sqrt{h^2 + (h+5)^2} \). Simplify and solve for \( d_1 \): \[ d_1 = \sqrt{h^2 + h^2 + 10h + 25} = \sqrt{2h^2 + 10h + 25} \].

Step 3: Calculate Diagonal of High-Definition Screen

Similarly, use the Pythagorean theorem for the high-definition screen: \( d_2 = \sqrt{h^2 + (1.8h)^2} \). Simplify and solve for \( d_2 \): \[ d_2 = \sqrt{h^2 + 3.24h^2} = \sqrt{4.24h^2} = \sqrt{4.24}h \].

Step 4: Set Up the Equation for Diagonal Difference

According to the problem, \( d_2 = d_1 + 14 \). Substitute the expressions from steps 2 and 3 to get:\[ \sqrt{4.24}h = \sqrt{2h^2 + 10h + 25} + 14 \].

Step 5: Solve the Equation for h

This equation can be solved by first isolating the square roots and squaring both sides to eliminate them. Then solve for \( h \). After algebraic manipulations, you will find the possible values for \( h \).

Step 6: Approximate Height to Nearest Tenth

After solving, find the positive value for \( h \) and round it to the nearest tenth as required by the problem, yielding the final solution.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a key mathematical concept used to determine the length of a side in a right-angled triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed in the formula:

• \( a^2 + b^2 = c^2 \)

where \( a \) and \( b \) are the lengths of the legs, and \( c \) is the length of the hypotenuse.
In the original exercise, the screen's rectangle uses the Pythagorean theorem to determine the diagonal (hypotenuse) based on its height and width. Understanding this concept allows one to find one side of a triangle when the lengths of the other two sides are known.
This theorem is not only fundamental in geometry but also widely applied in various fields such as architecture, physics, and engineering to solve practical problems involving distances.

Rectangle Geometry

Understanding the geometry of rectangles is crucial because these shapes are everywhere in real life. A rectangle is a four-sided flat shape where every angle is a right angle (90°). Its opposite sides are equal in length.
When solving problems involving rectangles, we often need to use their properties, such as:

• The length of each diagonal can be found using the Pythagorean theorem, since dividing a rectangle diagonally forms two right triangles.
• Perimeter, which is the total distance around the rectangle, is calculated as \( 2 \times (\text{width} + \text{height}) \).
• Area, the surface enclosed by the rectangle, is calculated as \( \text{width} \times \text{height} \).

In this exercise, each screen acts as a rectangle where the diagonal forms the hypotenuse of a right triangle with the width and height as the legs, allowing for the application of the Pythagorean theorem.

Equation Solving Steps

Solving equations step by step is like peeling an onion, one layer at a time. It involves isolating variables, simplifying expressions, and systematically working through mathematical operations.
In this exercise, the steps involve:

• Substituting known values and relationships, such as width in terms of height.
• Applying the Pythagorean theorem to express the diagonal in terms of height \( h \).
• Setting up an equation based on given conditions, here it's the difference in diagonals.
• Eliminating square roots by squaring both sides of the equation to simplify further.
• Solve the resulting polynomial equation for \( h \) and choose the viable solution based on context.
• Finally, round off to get the required precision.

Each step requires careful attention to detail and adherence to algebraic principles.

Measurement Conversion

Measurement conversion is an essential skill when dealing with dimensions and geometrical calculations. It involves changing a quantity expressed in one unit to another unit without altering the absolute quantity.
Though this exercise does not explicitly call for unit conversion, it's prudent to understand this skill in contexts where converting between units like inches, centimeters, or depending on the metric or imperial system, is necessary.
Key tips for effective measurement conversion include:

• Remembering common conversion factors, such as 1 inch = 2.54 centimeters.
• Applying conversion consistently across all related measurements to maintain accuracy.
• Being mindful of rounding, especially in scientific or technical contexts to maintain precision.

Whether determining screen sizes or scaling dimensions, conversion keeps calculations accurate in diverse applications.