Problem 35 - Technical Mathematics with Calculus

Question

Combined Operations with Approximate Numbers Perform each computation, keeping the proper number of digits in your answer. $$\left(\frac{8.90}{4.75}\right)^{2}$$

Step-by-Step Solution

Answer

The correct answer, rounded to three significant figures, is 3.50.

Step 1: Evaluate the Division

First, divide the number 8.90 by 4.75. To maintain proper significant figures, it's important to carry a sufficient number of decimal places during calculation.

Step 2: Square the Result

After obtaining the result from Step 1, square that number to find the final answer.

Step 3: Round to Proper Significant Figures

Since the numbers 8.90 and 4.75 each have three significant figures, the answer should also be rounded to three significant figures.

Key Concepts

Combined Operations

Combined operations in mathematical computation refer to the process where more than one arithmetic operations—such as addition, subtraction, multiplication, and division—are used in a sequence to solve an equation or an expression. When dealing with combined operations, particularly with approximate numbers, it is essential to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). It’s critical to perform each step with precision to avoid propagation of errors, which can significantly affect the final result. In practical applications, combined operations are everywhere, from computing financial interest to calculating physical forces.

Approximate Numbers

Approximate numbers are not exact, but close to the actual value. They are often the result of measurements or estimations and are essential in scientific computations where obtaining exact numbers is not always possible. When performing calculations with approximate numbers, it’s crucial to consider significant figures, which convey how precise these approximations are. The number of significant figures in a result should reflect the least precise measurement used in the computation to ensure that the result is neither overestimated nor underestimated in precision.

Rounding Significant Figures

Rounding to significant figures is a way to express a number to reflect its precision based on its least precise measurement. The rules of rounding help in maintaining the integrity of the data's precision during mathematical computations. To round a number to a desired number of significant figures:

• Identify the last significant figure that needs to be retained.
• Look at the next digit to the right; if it is 5 or more, increase the last retained digit by 1.
• Drop all digits to the right of the last retained digit.

Minding these rules in calculations is especially important in scientific and engineering contexts where precision is paramount.

Mathematical Computation

Mathematical computation involves performing arithmetic operations to obtain a numerical result. It is the fundamental process of math, enabling us to solve problems from the simplest addition to complex calculus. When engaging in mathematical computation with approximate numbers, precision is key. Carrying a sufficient number of decimal places during intermediate steps—not just the final result—and rounding only when the final answer is determined helps prevent the loss of accuracy. In the case of the given exercise, maintaining the proper number of decimal places throughout each computation step ensures the final answer is reliable and adheres to the principle of significant figures.