Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find the value of \(x\) to the nearest hundredth. $$ x e^{3}=e^{4} $$

Step-by-Step Solution

Answer

The value of \( x \) is approximately 2.72.

Step 1: Isolate the Variable

First, we need to isolate the variable \( x \) by dividing both sides of the equation by \( e^3 \). This yields:\[ x = \frac{e^4}{e^3} \]Recall that when dividing exponential terms with the same base, we subtract the exponents. Thus, \( \frac{e^4}{e^3} = e^{4-3} = e^1 \).

Step 2: Simplify the Expression

Now that we have simplified the exponent, we get:\[ x = e^1 \]Since \( e^1 = e \), we have:\[ x = e \]

Step 3: Evaluate \( e \)

To find the numerical value of \( e \) to the nearest hundredth, use the approximation \( e \approx 2.71828 \). Rounding this number to the nearest hundredth:\( e \approx 2.72 \).

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. In the equation provided, both sides contain exponential expressions with a base of Euler's number, denoted by \( e \). Euler's number is approximately equal to 2.71828 and is a fundamental constant in mathematics, especially seen in natural logarithms and growth calculations.

Here's what's happening in the exercise: we start with the equation \( x e^{3} = e^{4} \). The exponential expressions \( e^3 \) and \( e^4 \) feature the same base, making it easier to solve. Exponential functions like this one are crucial in fields such as compound interest calculations, population growth models, and various natural phenomena where changes multiply over time. Understanding how to manipulate and solve these functions is essential for both academic and real-world applications.

Isolating Variables

Isolating the variable is a crucial step when solving any equation, including those involving exponential functions. The main goal is to get the variable, in this case, \( x \), by itself on one side of the equation. In the original exercise, \( x \) is multiplied by \( e^3 \).

To isolate \( x \), we divide both sides of the equation by \( e^3 \), resulting in the expression \( x = \frac{e^4}{e^3} \). Dividing exponential expressions with the same base requires us to subtract the exponents: \( \frac{e^4}{e^3} = e^{4-3} \), which simplifies to \( e^1 \).

This method of handling exponential functions by isolating variables is essential for finding the actual value of unknowns in complex mathematical problems. Mastering this skill can significantly ease the process of solving similar equations.

Rounding Decimals

Rounding decimals is an important technique to make numbers more manageable and easier to read, especially when dealing with irrational numbers like \( e \). In the solution provided, once \( x \) is isolated and simplified to \( x = e \), we need a numerical approximation of \( e \).

The approximation given is \( e \approx 2.71828 \). However, to align with the exercise requirements, we need to round this to the nearest hundredth. This involves looking at the third digit after the decimal point. If it's 5 or greater, you'll round up the second digit by one. Thus, \( e \approx 2.72 \).

Rounding is useful in practical scenarios where precision is not just necessary but also needs to be balanced with simplicity. It's a basic yet vital skill in many quantitative fields, enabling clearer communication and understanding of numerical results.