Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 43-44: Approximate the real-number expression. Express the answer in scientific notation accurate to four significant figures. (a) \(\frac{1.2 \times 10^{3}}{3.1 \times 10^{2}+1.52 \times 10^{3}}\) (b) \(\left(1.23 \times 10^{-4}\right)+\sqrt{4.5 \times 10^{3}}\)

Step-by-Step Solution

Answer

(a) \(6.557 \times 10^{-1}\); (b) \(6.708 \times 10^{1}\).

Step 1: Solve the Denominator First

For equation (a), solve the expression in the denominator: \(3.1 \times 10^{2} + 1.52 \times 10^{3}\). Convert \(3.1 \times 10^{2}\) to \(310\), and \(1.52 \times 10^{3}\) is \(1520\). Add these together: \(310 + 1520 = 1830\.\)

Step 2: Evaluate the Quotient

Now solve \(\frac{1.2 \times 10^{3}}{1830}\). Calculate the numerator: \(1.2 \times 10^{3} = 1200\). Divide: \(\frac{1200}{1830} \approx 0.6557\.\)

Step 3: Express in Scientific Notation for (a)

Convert \(0.6557\) into scientific notation: \(6.557 \times 10^{-1}\). This is accurate to four significant figures as required.

Step 4: Calculate the Square Root

For equation (b), evaluate the square root: \(\sqrt{4.5 \times 10^{3}}\). Calculate \(\sqrt{4500} \approx 67.082\).

Step 5: Add the Two Parts Together

Now add the terms: \(1.23 \times 10^{-4} + 67.082\). Since \(1.23 \times 10^{-4} \approx 0.000123\), the sum is \(67.082 + 0.000123 \approx 67.082123\.\)

Step 6: Express in Scientific Notation for (b)

Convert \(67.082123\) to scientific notation, accurate to four significant figures: \(6.708 \times 10^{1}\).

Key Concepts

Understanding Approximation

Approximation is a mathematical technique used to find a number close enough to the correct answer without being exact. This makes complex calculations simpler and faster. In real-world scenarios, exact values are not always necessary; a close estimate can be sufficient. Approximations are common in areas like statistics and engineering, where speed and simplicity are prioritized over perfect precision.

• Approximations can save time.
• The result is not exact.

When we approximate, especially with decimals, we might round numbers to a certain number of places. This gives a result that is easy to understand but still very close to the true value. For example, calculating \( \frac{1200}{1830} \) results in about 0.655737... but for practical purposes, it is rounded to 0.6557 in calculations, which is easier to work with.

Significant Figures: Keeping Your Numbers Meaningful

Significant figures are crucial in scientific and engineering calculations, ensuring numbers retain only the relevant digits. This prevents overstatement of precision in measurements or calculations.

• Determines the digits that carry meaning.
• Avoids exaggerating precision.

In the context of our exercise, converting numbers into four significant figures ensures uniformity and clarity in results. For instance, 0.6557 becomes significant at the fourth decimal place, thus shown in scientific notation as \(6.557 \times 10^{-1}\). Each digit here contributes to the accuracy of the solution, reinforcing the importance of understanding their impact on the final output.

Square Root Calculation Demystified

Calculating square roots, especially of large numbers, can initially seem daunting. However, this operation can be simplified by breaking the number into more manageable factors—making the process less intimidating. When tasked with \(\sqrt{4.5 \times 10^3}\), understanding the problem becomes clearer if you simplify it to the root of 4500.

• Break down the number: such as \(4500\) becomes \(45 \times 10^2\).
• Calculate the square roots separately: \(\sqrt{45} \approx 6.71\) and \(\sqrt{10^2} = 10\), thus \(\sqrt{4500} \approx 67.1\).

Once broken down, find the square roots individually before combining them back. In practical terms, approximations like these follow logical steps to produce an easily understandable result—for example, the answer in scientific notation, herein, shows the clarity of this operation.