Problem 37 - Intermediate Algebra

Question

e-Readers. The perimeter of the rectangular display screen of Amazon's Kindle 3 is 16.8 inches. If the height is 1.2 inches greater than the width, find the dimensions of the screen. (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The dimensions of the screen are 3.6 inches in width and 4.8 inches in height.

Step 1: Understand the Problem

We are given that the perimeter of the rectangular screen is 16.8 inches. We know that the height is 1.2 inches greater than the width. We need to find the width and height of the screen.

Step 2: Write the Formula for Perimeter

The formula for the perimeter of a rectangle is given by \( P = 2(l + w) \), where \( l \) is the length, in this case the height, and \( w \) is the width.

Step 3: Express Height in Terms of Width

Since the height is 1.2 inches greater than the width, we can express the height as \( h = w + 1.2 \).

Step 4: Substitute and Set Up the Equation

Substitute \( h = w + 1.2 \) into the perimeter formula: \[ 2(w + (w + 1.2)) = 16.8 \] This simplifies to: \[ 2(2w + 1.2) = 16.8 \] Then further to: \[ 4w + 2.4 = 16.8 \]

Step 5: Solve for Width

Subtract 2.4 from both sides of the equation: \[ 4w = 16.8 - 2.4 \] \[ 4w = 14.4 \] Now, divide both sides by 4: \[ w = \frac{14.4}{4} = 3.6 \] Thus, the width is 3.6 inches.

Step 6: Find the Height

Using the expression for the height: \[ h = w + 1.2 = 3.6 + 1.2 = 4.8 \] The height is therefore 4.8 inches.

Key Concepts

Rectangles

A rectangle is a fundamental shape in geometry and algebra characterized by having four sides and four right angles. The opposite sides of a rectangle are equal, meaning they have the same length. This unique feature allows us to see rectangles not just as shapes, but as a basis for solving various mathematical problems involving area, perimeter, and more. Rectangles are used widely in everyday objects like books, screens, and rooms.

For our specific problem, we deal with a rectangular display screen where the relationship between the width and height needs to be determined. Understanding these basic properties of a rectangle helps in applying mathematical formulas effectively to solve such problems.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Systems of equations can be solved using various methods, such as substitution, elimination, and graphical methods. In context with the problem we are dealing with, we use the substitution method.

The idea here is simple. We know two things about the display screen: the perimeter and the relationship between the height and width.

• The perimeter is 16.8 inches.
• Height is 1.2 inches greater than the width.

By expressing the height in terms of the width using the equation "height = width + 1.2," we can substitute this into the perimeter formula to solve for the width. Once the width is known, we can easily find the height. Utilizing a system of equations provides a logical flow to tackling such interrelated measurements in math.

Perimeter

Perimeter is the total distance around a two-dimensional shape, in this case, a rectangle. The formula to calculate the perimeter of a rectangle is crucial for many real-life applications.

For a rectangle, the perimeter is given by the formula:

• \( P = 2(l + w) \)

where \( l \) stands for the length (or height in the context of the problem), and \( w \) represents the width. Knowing this formula enables you to calculate the perimeter when dimensions are known or vice versa, which is what happens in our exercise.

To solve problems involving perimeter, it is important to understand that you're essentially adding the lengths of all sides. This exercise involved dealing with given perimeter and using it to discover dimensions by re-arranging the common perimeter formula, proving just how versatile knowledge of this calculation is.