Problem 55 - Calculus Early Transcendentals: Pearson New International Edition

Question

find the best decimal approximation that your calculator allows. Begin by making a mental estimate $$ \sqrt{8.9 \pi^{2}+1}-3 \pi $$

Step-by-Step Solution

Answer

The best decimal approximation is approximately 0.032.

Step 1: Identify the Expression

The given expression is \( \sqrt{8.9 \pi^{2}+1}-3 \pi \). We will evaluate this step by step to find the best decimal approximation.

Step 2: Estimation of Terms

Before using a calculator, make a mental estimate of the values of the constants involved. We know that \( \pi \approx 3.14 \), so \( \pi^2 \approx 9.86 \). Calculate roughly: \( 8.9 \times 9.86 \approx 87.65 \). Add 1 to get \( 88.65 \). Thus, \( \sqrt{88.65} \) should be approximated.

Step 3: Calculate \( \sqrt{8.9 \pi^2 + 1} \)

Use your calculator to find a more precise value of \( \sqrt{8.9 \pi^{2}+1} \). With the exact value of \( \pi \approx 3.14159 \): \( 8.9 \times (3.14159)^2 + 1 \approx 89.4217 \). So, \( \sqrt{89.4217} \approx 9.457 \).

Step 4: Subtract \( 3\pi \)

Calculate \( 3 \pi \) using the approximation \( \pi \approx 3.14159 \). So, \( 3 \pi \approx 3 \times 3.14159 = 9.42477 \). Subtract this from \( 9.457 \) to get the final result: \( 9.457 - 9.42477 \approx 0.03223 \).

Step 5: Final Verification

Check the calculations using the calculator to ensure the approximation is precise. Verifying each significant figure will ensure correctness. The calculated value should match the final result approximately \( 0.032 \).

Key Concepts

Decimal Approximation

Decimal approximation is an essential mathematical tool that helps us simplify complex calculations to a form that’s easier to work with. In this exercise, we aim to find a decimal approximation of the expression \( \sqrt{8.9 \pi^{2} + 1} - 3 \pi \). Achieving a precise value isn't always necessary; a quick estimate can often suffice to give us an idea of the magnitude of the answer. Approximations are particularly useful when we don’t need the exact number but an estimated value for further calculations or if the exact number is cumbersome to handle. To apply a decimal approximation, start by identifying important constants like \( \pi \). Knowing that \( \pi \approx 3.14 \), and each term in our expression, we can simplify our calculations. Approximations are best used when manually verifying results with fractions and operations that produce irrational numbers like square roots.

Mathematical Estimation

Mathematical estimation allows us to make educated guesses about the size of an answer without calculating it exactly. This is helpful in everyday math problems and complex expressions. In our problem, before diving into calculations, we make a mental approximation:

• Estimate \( \pi \) as \( 3.14 \).
• Use simple calculations to determine \( 8.9 \times 9.86 \approximates 87.65 \).
• Add 1 for \( 88.65 \).

When estimating, it's key to start with known values or values close to it. It simplifies the complexity and gives a ballpark figure for validation against precise calculations. Such practice sharpens your ability to deal with numbers swiftly and efficiently even without a calculator.

Calculator Use

Calculators are invaluable tools in modern mathematics for quickly crunching numbers that are tedious or difficult to calculate by hand. In our expression, we used a calculator to verify our mental estimate and get a more precise value. To effectively use your calculator, ensure you input values as accurately as possible. In this case:

• Use the calculator to compute \( 8.9 \times (3.14159)^2 + 1 \) for precision.
• Calculate the square root directly with your device, like \( \sqrt{89.4217} \).
• Follow through with subtraction, \( 9.457 - 9.42477 \).

Accuracy in inputting each number is critical for calculators, especially when dealing with powers and roots that can easily throw off your result if handled improperly. Double-checking with a calculator also confirms that your mental estimates are on track.

Square Root Calculation

Square root calculation is often encountered in algebra and calculus problems. It involves finding a number that, when multiplied by itself, gives the original number. In this problem, after simplifying inside the square root, we find \( \sqrt{89.4217} \). Calculating a square root can occasionally be estimated first and then refined:

• Realize \( \sqrt{88.65} \) approximately equals 9.4 by rough estimation.
• The precise calculation gives \( \sqrt{89.4217} \approx 9.457 \).

Knowing how to handle square roots, especially with long decimals, is handy for both precise and quick-statement answers. It strengthens problem-solving techniques you learn and apply in advanced math concepts.