Problem 60 - Algebra for College Students

Question

A square pixel on a computer screen has a side of length (1.17) \(\left(10^{-2}\right)\) inches. Find the approximate area of the pixel in inches. Express the result in decimal form.

Step-by-Step Solution

Answer

The area of the pixel is approximately 0.00013689 square inches.

Step 1: Understanding the Problem

We are given the side length of a square pixel as \(1.17 \times 10^{-2}\) inches. We need to find the area of the pixel. Since it's a square, the area will be the side length squared.

Step 2: Calculating the Square of the Side

The formula for the area of a square is \(A = s^2\), where \(s\) is the side length. Thus, the area of the pixel is \((1.17 \times 10^{-2})^2\). We will calculate this next.

Step 3: Simplifying the Expression

First, calculate the square of the side: \((1.17)^2 = 1.3689\). Next, calculate the square of \(10^{-2}\): \((10^{-2})^2 = 10^{-4}\).

Step 4: Multiplying the Components

Combine the results of the squared side and power of ten: \(1.3689 \times 10^{-4}\). This is the area of the square in square inches.

Step 5: Expressing the Final Answer in Decimal Form

Convert \(1.3689 \times 10^{-4}\) into a regular decimal form: \(0.00013689\). This is the area of the pixel in square inches.

Key Concepts

Square Units

When we talk about the area of a shape, we usually refer to square units. Square units are a way to measure how much space a two-dimensional shape covers. For example, when you are calculating the area of a square, you count how many of these square units fit inside it.

In the case of our square pixel, its area is given in square inches. Remember, a square inch is not a size but a measurement that shows how much surface a figure contains. To find the area, we take the side of the square and multiply it by itself.

• If \(s\) represents the side of the square, the area \(A\) will be \(s^2\).
• The square pixel has a side length of \(1.17 \times 10^{-2}\ inches\)—a small measurement because pixels are tiny parts of a screen.

Understanding square units helps make sense of how areas represent the size of spaces and surfaces.

Decimal Form

Numbers come in many forms and one common representation is the decimal form. This form is especially useful when we need to present precise measurements or very small numbers.

In our problem, we convert a number from scientific notation (1.3689 multiplied by \(10^{-4}\)) into standard decimal form, which simplifies it and makes it easier to read.

• The power of \(10^{-4}\) suggests that we move the decimal point four places to the left.
• Thus, the conversion of \(1.3689 \times 10^{-4}\) results in \(0.00013689\).

Decimal form is crucial in everyday math and science because it shows precise values, allowing for clear communication and understanding.

Algebraic Calculations

Solving problems involving areas often requires algebraic calculations. It includes operations like squaring numbers, simplifying expressions, and converting between different forms.

For the given pixel, we started by squaring its side length. To calculate this:

• If given \(s = 1.17 \times 10^{-2}\), determine \(s^2 = (1.17 \times 10^{-2})^2\).
• Calculate separately: \((1.17)^2 = 1.3689\) and \((10^{-2})^2 = 10^{-4}\).

The critical step is combining these results: multiply \(1.3689 \times 10^{-4}\), which gives the area of the pixel.

Understanding these operations helps to solve problems and makes algebra a valuable tool in mathematics. Practicing with these helps improve speed and accuracy in calculations.