Problem 42 - Technical Mathematics with Calculus

Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\sqrt{9.74}+\sqrt{12.5}$$

Step-by-Step Solution

Answer

\(\sqrt{9.74} + \sqrt{12.5} \approx 6.66\)

Step 1: Estimate Square Roots

Estimate the square root of each number to the nearest hundredth. Use a calculator if necessary: - For \(\sqrt{9.74}\), since \(9 < 9.74 < 16\), the square root should be between 3 and 4. Calculating gives \(\sqrt{9.74} \approx 3.12\).- For \(\sqrt{12.5}\), since \(9 < 12.5 < 16\), the square root should also be between 3 and 4. Calculating gives \(\sqrt{12.5} \approx 3.54\).

Step 2: Perform the Addition

Add the estimated square roots to get the approximate sum: \[3.12 + 3.54 = 6.66\]

Step 3: Consider Significant Figures

Since we were asked to keep the proper number of digits, and our initial approximations for the square roots were to the nearest hundredth, our final answer should also be reported to the nearest hundredth, which is already done in Step 2.

Key Concepts

Square Root Estimation

Understanding Square Root Estimation is essential when solving problems involving square roots, especially when they create part of a bigger equation or set of operations. The estimation of a square root involves finding an approximate value that comes close to the actual square root. One common method involves identifying the perfect squares closest to the number you are estimating. For example, given the task to estimate \(\sqrt{9.74}\), it is acknowledged that the perfect squares closest to 9.74 are 9 (\(3^2\)) and 16 (\(4^2\)). This suggests that the square root of 9.74 should lie between 3 and 4.

Using a calculator or estimation strategies, you can determine a more precise approximation. If you're estimating without a calculator, it's helpful to consider the distance of your number from the nearest perfect squares to guide your estimation. If \(9.74\) is closer to 9, then the estimated value will be closer to 3 than to 4. In this case, calculating gives an estimation of approximately 3.12, which serves as a reasonable starting point for further operations.

Significant Figures

The concept of Significant Figures is pivotal in mathematics and science since it deals with the precision of numbers. Significant figures are the digits in a number that carry meaning contributing to its precision. This includes all the numbers from the first non-zero digit to the last non-zero digit or any final zeros to the right of the decimal point. For instance, in an approximation task such as \(\sqrt{9.74} + \sqrt{12.5}\), if each square root is estimated to the nearest hundredth, the precision is up to two decimal places.

When combining these numbers in further operations, it's important to maintain this level of precision throughout — not doing so could result in an incorrect answer. In our example, the final sum, calculated as 6.66, preserves the hundredth place, ensuring that the significant figures from the estimated roots are properly carried through to the result. This reflects the proper application of significant figures in addition.

Combined Operations

Working with Combined Operations involves undertaking more than one arithmetic operation, such as addition, subtraction, multiplication, or division, in a sequence. In problems with approximate numbers, precision remains paramount through each step. For example, let's examine combining the estimated square roots from our previous section's problem. After estimating \(\sqrt{9.74}\) as approximately 3.12 and \(\sqrt{12.5}\) as approximately 3.54, these two values must be added.

For the addition \(3.12 + 3.54\), it is simple arithmetic yielding 6.66. Notably, in combined operations, maintaining the correct number of significant figures throughout the process is vital to ensure the result's accuracy. If we were to round off too early or not follow the rules of significant figures, it could lead to a less accurate outcome. Furthermore, understanding the order of operations is crucial if the problem includes a mix of addition, multiplication, and other operations; calculations should be performed in the correct sequence to achieve the correct result.