Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x=e^{1.5} $$

Step-by-Step Solution

Answer

The value of \( x \) is approximately 4.48.

Step 1: Understanding the Problem

Our task is to find the value of \( x \) given as \( x = e^{1.5} \). \( e \) is a mathematical constant approximately equal to 2.71828, and the expression involves raising \( e \) to the power of 1.5.

Step 2: Evaluate the Exponential Expression

To find \( x = e^{1.5} \), calculate \( e \) raised to the power of 1.5. This can be done using a scientific calculator or evaluation software.

Step 3: Calculation Process

Using a calculator, input \( e^{1.5} \) to get approximately 4.48169.

Step 4: Rounding to the Nearest Hundredth

Rounding 4.48169 to the nearest hundredth means looking at the third decimal place (1 in this case). Since this is less than 5, we round down to get 4.48.

Key Concepts

Understanding Euler's Number

Euler's Number, often denoted by the letter \( e \), is a fundamental mathematical constant. It is similar in importance to the number \( \pi \) in the realm of natural phenomena mathematics.
Unlike physical constants, Euler's Number is about growth in processes like compounding interest, population growth, and radioactive decay. Its value is approximately \( 2.71828 \).

• \( e \) arises naturally in problems involving exponential growth or decay.
• It is the base of natural logarithms, meaning the logarithm of \( e \) itself is 1.

When you see an expression like \( e^{1.5} \), understanding Euler's number helps comprehend how such growth or multiplication happens exponentially. Thus, every increase on the exponent implies significant growth of the whole number.

Rounding Numbers Made Easy

Rounding numbers is a crucial skill in math that helps simplify figures without significantly losing accuracy. It's about deciding how many digits are appropriate for a given situation.
The process involves looking at the number right after the place you wish to round to. For instance, rounding to the nearest hundredth means focusing on the thousandths digit.

• If the thousandths digit is 5 or more, you round up the hundredths digit by one.
• If it's less than 5, you keep the hundredths digit as it is.

In our example, we rounded 4.48169 to 4.48 because the third digit (1) is less than 5. Always take care when rounding, as every decimal place holds value depending on context.

Using a Scientific Calculator Effectively

A scientific calculator is a handy tool when dealing with complex computations, including exponential expressions. Here are steps to get the best of it when calculating powers of \( e \):
First, ensure your calculator has an \("e^x"\) function or similar. Most scientific calculators do because of Euler's Number's importance in advanced math calculations.

• Type \( 1.5 \) into the calculator.
• Press the \("e^x"\) or equivalent function.
• Your calculator should display the result of the exponential operation.

Remember, practice will improve your speed and confidence. Knowing how to use your calculator effectively means you do not just rely on what it tells you, but also understand why it shows a specific number.