Problem 70 - A Graphical Approach to College Algebra

Question

Approximate each expression to the nearest hundredth. $$2 \pi^{3}-5 \pi^{2}-3$$

Step-by-Step Solution

Answer

The approximate value to the nearest hundredth is 9.66.

Step 1: Understanding the Expression

The expression we need to approximate is \(2 \pi^{3} - 5 \pi^{2} - 3\). This expression involves powers of \(\pi\), which is approximately 3.14159. Our goal is to compute the values of \(\pi^{3}\) and \(\pi^{2}\) first.

Step 2: Calculating \(\pi^{2}\)

Calculate \(\pi^{2} = 3.14159^{2}\). Use a calculator to find this value precisely. After calculation, \(\pi^{2} \approx 9.8696\).

Step 3: Calculating \(\pi^{3}\)

Calculate \(\pi^{3} = 3.14159 \times 9.8696\). Use a calculator to find this result. The calculated value is \(\pi^{3} \approx 31.0062\).

Step 4: Substituting Back into the Expression

Now substitute the calculated values of \(\pi^{2}\) and \(\pi^{3}\) back into the original expression: \(2 \times 31.0062 - 5 \times 9.8696 - 3\).

Step 5: Calculating Each Term in the Expression

Compute each term separately: \(2 \times 31.0062 = 62.0124\), \(5 \times 9.8696 = 49.348\).

Step 6: Combining the Terms

Combine the results: \(62.0124 - 49.348 - 3\). Simplify this calculation: \(62.0124 - 49.348 = 12.6644\), then \(12.6644 - 3 = 9.6644\).

Step 7: Rounding to Two Decimal Places

Finally, round the result \(9.6644\) to the nearest two decimal places, or hundredths. Thus, the final answer is approximately \(9.66\).

Key Concepts

Arithmetic Operations

Arithmetic operations are the basic calculations you perform with numbers, including addition, subtraction, multiplication, and division. In algebra, solving expressions often involves combining these operations to simplify or solve the expression. Understanding the order of these operations is crucial.

• Addition and Subtraction: These operations are about combining numbers together. Remember that subtraction is the reverse or inverse of addition.
• Multiplication and Division: Multiplication is repeated addition, while division undoes multiplication. These are used to scale numbers up or down.
• Order of Operations: Follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) to solve expressions in the correct sequence.

In the expression given, you see multiplication and subtraction. Each term is computed separately, respecting the order of operations. First, calculate powers and products, then perform subtraction.

Pi (π) Calculations

Pi (\(\pi\)) is a special number approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter. Due to its transcendental nature, Pi is an infinite non-repeating decimal.When calculating expressions involving Pi, it's important to know how to work with its approximations:

• Square of Pi: Calculating \(\pi^2\) is simply \(3.14159^2 = 9.8696\). This means multiplying Pi by itself once.
• Cube of Pi: For \(\pi^3\), multiply Pi by \(\pi^2\) or \(3.14159 \times 9.8696 = 31.0062\). It shows how Pi behaves when taken to higher powers.

In practical calculations, using calculators helps handle Pi's infinite decimals, achieving the needed precision, such as in this exercise.

Rounding Decimals

Rounding decimals means reducing the number of decimal places in a number while keeping its value close to the original. It involves looking at the place value to which you want to round and making decisions based on the digit that follows.For rounding to the hundredth place:

• Identify the Place: The hundredth place is the second digit after the decimal point.
• Check the Digit: Look at the third digit. If it is 5 or greater, round the second digit up. If it's less than 5, keep the second digit as is.

For example, in the result \(9.6644\), the third digit 4 makes the hundredth digit 6 remain unchanged, therefore, it rounds to \(9.66\). This step simplifies numbers for easier interpretation and comparison.