Problem 27 - College Algebra with Modeling and Visualization

Question

Write a system of linear equations with two variables whose solution satisfies the problem. State what each variable represents. Then solve the system. The screen of a rectangular television set is 2 inches wider than it is high. If the perimeter of the screen is 38 inches, find its dimensions.

Step-by-Step Solution

Answer

The television screen is 8.5 inches high and 10.5 inches wide.

Step 1: Define variables

Let the height of the television screen be denoted by \( h \), and the width of the screen be represented by \( w \).

Step 2: Set up equations

We know from the problem statement that the width is 2 inches wider than the height. Hence, we have:\[w = h + 2\] Also, the perimeter of a rectangle is given by \( 2 \times \text{(height + width)} = 38 \). This gives us the equation:\[2(h + w) = 38\]

Step 3: Simplify perimeter equation

Divide the perimeter equation by 2 to simplify it:\[h + w = 19\]

Step 4: Substitute and solve for height

Substitute \( w = h + 2 \) into the simplified perimeter equation \( h + w = 19 \):\[h + (h + 2) = 19\]Combine terms:\[2h + 2 = 19\]Subtract 2 from both sides:\[2h = 17\]Divide by 2 to solve for \( h \):\[h = 8.5\]

Step 5: Solve for width

Use the expression for width: \( w = h + 2 \). Substitute \( h = 8.5 \):\[w = 8.5 + 2 = 10.5\]

Step 6: Conclusion

The height of the television screen is 8.5 inches, and the width is 10.5 inches. These dimensions satisfy both the given perimeter and the condition that the width is 2 inches greater than the height.

Key Concepts

Rectangular Dimensions

In geometry, rectangles are shapes with four sides and four right angles. Understanding their dimensions is fundamental. A rectangle's dimensions refer to its length and width, typically the longest and shortest sides respectively when oriented in a standard view. Both dimensions determine its size and area. Here, we look at practical applications of these dimensions. Rectangles appear in various real-life applications, such as television screens and picture frames. For example, a television screen with specific rectangular dimensions fits certain spaces or displays content clearly. A key aspect of rectangles is that opposite sides are equal, making calculations easier. When addressing problems involving rectangles, it's important to note all parameters including the relationship between the sides, and how these could affect other properties like the perimeter.

Perimeter Calculation

Calculating the perimeter of a rectangle is a straightforward process, yet it's crucial for determining the border dimensions. The perimeter is the total distance around the rectangle, calculated by adding up all its sides. For a rectangle, the formula is:

• Perimeter = 2 × (Width + Height)

This formula stems from the idea that a rectangle has two widths and two heights. In the problem provided, the perimeter is set at 38 inches. By using the perimeter formula, we derived an equation that helps find the dimensions of the rectangle when given certain conditions such as one dimension being longer than the other. Additionally, solving the perimeter equation often involves simple arithmetic operations like addition and multiplication. By understanding this, we can solve complex problems efficiently.

Variable Representation

When dealing with mathematical problems, it's essential to represent unknown quantities with variables. This approach helps organize data and find solutions logically. In our rectangle problem, we use variables to denote the height and width of the television screen.Variables are placeholders for numeric values and are represented by symbols such as letters. We used:

• \( h \): for the height of the rectangle.
• \( w \): for the width of the rectangle.

This specific representation lets us form equations based on the problem's details.For example, the given condition that the width is 2 inches more than the height is translated into an equation: \( w = h + 2 \). This aids in substituting and simplifying problems involving systems of equations.In solving systems of equations, it’s important to represent relationships between variables clearly. This provides clarity and aids in reaching accurate solutions.