Problem 13 - A Graphical Approach to College Algebra

Question

$$\begin{aligned} &\text {Solve each problem analytically, and support your solution}\\\ &\text {graphically.} \end{aligned}$$ The aspect ratio of older television monitors is \(4: 3 .\) One such television has a rectangular viewing screen with perimeter 98 inches. What are the length and width of the screen? since televisions are advertised by the diagonal measure of their screens, how would this monitor be advertised? (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

Dimensions are 28 inches by 21 inches, advertised as a 35-inch TV.

Step 1: Understand the Problem

We need to find the dimensions of a television screen with a perimeter of 98 inches and an aspect ratio of 4:3. The length and width will correspond to these dimensions. Finally, calculate the diagonal to determine how the TV would be advertised.

Step 2: Setup the Equations

Given the aspect ratio 4:3, let the width be \(4x\) and the height be \(3x\). The perimeter \(P\) of a rectangle is given by \(P = 2(l + w)\). Therefore, for this TV, we have the equation: \[ 2(4x + 3x) = 98 \].

Step 3: Solve for x

Simplify the equation: \[ 2(7x) = 98\] \[ 14x = 98 \]. Solving for \(x\) gives: \[ x = \frac{98}{14} = 7 \].

Step 4: Calculate Length and Width

Using \(x = 7\), calculate the width and height using the aspect ratio dimensions: - Width: \(4x = 4 \times 7 = 28\) inches. - Height: \(3x = 3 \times 7 = 21\) inches. Thus, the dimensions are 28 inches by 21 inches.

Step 5: Determine the Diagonal

The diagonal \(d\) of a rectangle is given by the Pythagorean theorem: \[ d = \sqrt{l^2 + w^2} \]. Substitute the values: \[ d = \sqrt{28^2 + 21^2} = \sqrt{784 + 441} = \sqrt{1225} = 35 \].

Step 6: Interpret the Results

The TV has dimensions 28 inches by 21 inches, and the diagonal is 35 inches. This means the TV would be advertised as a "35-inch TV" due to its diagonal measurement.

Key Concepts

Aspect Ratio

The aspect ratio is a term often used to describe the proportional relationship between the width and height of an object, such as a television screen or a photograph. In the context of television screens, aspect ratios indicate how wide the screen is in relation to its height.

For older television monitors, the aspect ratio is set at 4:3. This means for every 4 units of width, there are 3 units of height. If you think of a scaled-down version, for every 4 inches wide, the screen would be 3 inches tall.

This fixed ratio helps maintain the integrity of the display image, ensuring images appear naturally without distortion. Aspect ratios are crucial in both designing screens and in content creation because they affect how visuals are presented.

Rectangle Perimeter

The perimeter of a rectangle is the total distance around its outer ed ges. It's calculated by adding up the lengths of all four sides. In formula form, it looks like this: \[ P = 2(l + w) \] where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width.

In this exercise, the given perimeter was 98 inches. For any rectangle with this perimeter, understanding the relationship between length and width is key to solving for physical dimensions when combined with an aspect ratio.

When solving the problem, we set up an equation using the rectangle's perimeter: first multiplying the sum of its length and width by two. In this case, the dimensions maintain the 4:3 aspect ratio. This means if width is expressed in terms of a variable, say \(x\), then length could be expressed in terms of another multiple, such as \(3x\). This approach simplifies calculations and helps uncover the real-life dimensions of screens.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry, particularly useful in calculating the length of the diagonal in a right-angle triangle or rectangle. It states that in a right-angled triangle, the square of the length of the hypotenuse \((c)\) is equal to the sum of the squares of the other two sides \((a\) and \(b)\): \[ c^2 = a^2 + b^2 \]

In the case of our rectangular TV screen, the diagonal forms the hypotenuse, while the length and width form the other two sides. By applying the Pythagorean theorem, we calculated the diagonal (important for how TVs are marketed).

Using the dimensions derived from the perimeter and the aspect ratio (28 inches for length and 21 inches for height), we plugged these into the theorem: \[ d = \sqrt{28^2 + 21^2} \] Ultimately, this led us to a 35-inch diagonal. The Pythagorean theorem, thus, transforms abstract measurements into tangible results, which in this case tells us how the TV would be advertised based on its diagonal measurement.