Problem 59 - Intermediate Algebra

Question

For Problems \(54-61\), perform the following calculations and express answers to the nearest hundredth. (These calculations are in preparation for our work in the next section.) $$ \frac{\ln 2}{0.03} $$

Step-by-Step Solution

Answer

The answer is 23.10.

Step 1: Calculate the Natural Logarithm

Identify the logarithm part of the expression. In this problem, we need to calculate \( \ln 2 \). Using a calculator or logarithm table, find \( \ln 2 \approx 0.6931 \).

Step 2: Perform the Division

Take the result from Step 1 and divide it by 0.03. \[ \frac{0.6931}{0.03} = 23.1033 \]

Step 3: Round the Answer

Round the result from Step 2 to the nearest hundredth as required. \( 23.1033 \) rounded to the nearest hundredth is \( 23.10 \).

Key Concepts

Understanding Natural Logarithms

Natural logarithms are a fundamental concept in mathematics, often represented using the notation \( \ln \). This type of logarithm uses the base \( e \), where \( e \) is approximately equal to 2.71828. The natural logarithm of a number tells us the power to which \( e \) has to be raised to produce that number. This can be quite abstract, but it becomes more manageable when we use tools like calculators to find these values. For example, when calculating \( \ln 2 \), you can use a calculator to find the value is approximately 0.6931.

• Natural logarithms are used extensively in calculus and real-world phenomena like growth and decay.
• The constant \( e \) is a fundamental part of natural logarithms and appears in many areas of mathematics and science.

Understanding natural logarithms is critical because they help simplify complex calculations, especially those involving exponential growth or decay.

Rounding Numbers to the Nearest Hundredth

Rounding numbers is an essential skill in mathematics, especially when precision is crucial, but not overly necessary. When you round to the nearest hundredth, you look at the decimal third place. If the digit in the third decimal place is 5 or greater, round the second decimal place up by one. If it's less than 5, keep the second decimal place unchanged. For instance, after performing a division, you might get the result 23.1033. Here is how you would round: - Identify the number in the third decimal place, which is 3 in our case. - Since 3 is less than 5, we round down, keeping the number in the second decimal place unchanged. - Thus, 23.1033 rounded to the nearest hundredth becomes 23.10. Rounding is important to maintain simplicity and manageability in your calculations without losing significant detail.

Performing Division Calculations

Division is one of the basic arithmetic operations, and it involves finding how many times one number is contained within another. When dealing with more complex mathematical expressions, like the one involving natural logarithms, division is a necessary step to simplify or solve an equation.In the example \( \frac{0.6931}{0.03} \), the operation signifies dividing the natural logarithm of 2 by 0.03. You can perform this calculation using a calculator or by manual division. Here's a simplified approach:

• Think of division as splitting a number into equal parts or groups.
• For \( \frac{0.6931}{0.03} \), ask: "how many 0.03s fit into 0.6931?"

A calculator typically makes quick work of this, giving you approximately 23.1033, which you would then round. Division is crucial for solving equations and understanding proportions between numbers.