Problem 13 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. Aspect Ratio of a Television Monitor 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? (GRAPH CAN'T COPY)

Step-by-Step Solution

Answer

Width: 28 inches, Height: 21 inches, Diagonal (advertised size): 35 inches.

Step 1: Define Variables

Let the width of the screen be \(4x\) and the height be \(3x\). We chose these expressions because the aspect ratio is 4:3, meaning for every 4 units of width, there are 3 units of height.

Step 2: Use Perimeter Formula

The perimeter of the rectangle is given as 98 inches. The formula for the perimeter of a rectangle is \(P = 2(l + w)\). Substitute the length and width related to \(x\): \(98 = 2(4x + 3x)\).

Step 3: Solve for x

Simplify the equation from Step 2: \(98 = 14x\). Solve for \(x\) by dividing both sides by 14: \(x = 7\).

Step 4: Find Length and Width

Now that we have \(x = 7\), substitute back to get the actual dimensions: \( \) Width = \(4x = 28\) inches, \( \) Height = \(3x = 21\) inches.

Step 5: Calculate Diagonal

Use the Pythagorean theorem to find the diagonal size, \(d\), since the screen can be considered as a right triangle: \(d = \sqrt{(28)^2 + (21)^2}\). Calculate: \(d = \sqrt{784 + 441} = \sqrt{1225} = 35\) inches.

Step 6: Draw a Graph

Even though we cannot copy the graph here, you can represent these values graphically by drawing a rectangle with dimensions 28 x 21 on graph paper, and then draw a diagonal line from one corner to the opposite corner to visualize the 35-inch diagonal.

Key Concepts

Rectangle Perimeter

The perimeter of a rectangle is a fundamental concept in geometry that helps us determine the total length around a rectangular shape. For any rectangle, the perimeter (\(P\)) is calculated using the formula:

• \(P = 2(l + w)\)

where \(l\) is the length and \(w\) is the width.
In our problem, the television screen has a perimeter of 98 inches. Replacing \(l\) and \(w\) with expressions involving the variable \(x\) derived from the aspect ratio (4:3), \(l\) is expressed as \(4x\) and \(w\) as \(3x\).
This transforms the perimeter formula into:

• \(98 = 2(4x + 3x)\)

This equation simplifies and allows us to find the value of \(x\), leading to the actual dimensions of the screen.

Pythagorean Theorem

The Pythagorean Theorem is an essential mathematical tool used in determining the length of the sides of right triangles. Formulated as:

• \(a^2 + b^2 = c^2\)

where \(a\) and \(b\) are the lengths of the legs and \(c\) is the hypotenuse.
In the context of our television screen, the screen's width and height form the two legs of a right triangle, and the diagonal is the hypotenuse.
Given the screen's width of 28 inches (\(a\)) and height of 21 inches (\(b\)), we apply the theorem to find the diagonal (\(c\)):

• \(c = \sqrt{28^2 + 21^2} = \sqrt{784 + 441} = \sqrt{1225} = 35\)

This calculation reveals that the diagonal of the screen, which is crucial for advertisement purposes, measures 35 inches.

Analytical Solution

An analytical solution involves solving a problem using algebraic, logical, and systematic approaches without necessarily relying solely on numerical estimation or trial and error.
In this scenario, the use of algebra helps us determine the original dimensions of the television screen given a fixed perimeter and aspect ratio. By setting expressions for the width (\(4x\)) and height (\(3x\)) relative to the variable \(x\), we can input these values into the perimeter equation, simplify, and solve for \(x\).
After identifying \(x\), substituting back gives us explicit dimensions, confirming the television's screen measurements:

• Width: \(28\) inches
• Height: \(21\) inches

We then employ the Pythagorean Theorem to analytically confirm the diagonal length. Such a systematic approach ensures accuracy and clear understanding, allowing us to describe this process as solving the problem analytically.