Problem 18 - Elementary and Intermediate Algebra

Question

Let \(f(x)=75+3,570 \ln x .\) Find \(f(28.1) .\) Round to the nearest tenth.

Step-by-Step Solution

Answer

The value of \( f(28.1) \) is approximately 11,992.4.

Step 1: Identify the Function and Input Value

We are given the function \( f(x) = 75 + 3,570 \ln x \) and are asked to find \( f(28.1) \). Our task is to substitute \( x = 28.1 \) into the function and evaluate it.

Step 2: Substitute the Input Value into the Function

Replace \( x \) with 28.1 in the expression for \( f(x) \). We get: \[ f(28.1) = 75 + 3,570 \ln(28.1) \].

Step 3: Calculate the Natural Logarithm

Use a calculator to find the natural logarithm of 28.1: \( \ln(28.1) \approx 3.33656 \).

Step 4: Multiply the Natural Logarithm by 3,570

Compute \( 3,570 \times 3.33656 \) to get \( 11,917.4292 \).

Step 5: Add the Constant Term

Now, add 75 to the result of the previous step: \( 11,917.4292 + 75 = 11,992.4292 \).

Step 6: Round the Result

Round the result from Step 5 to the nearest tenth, which gives us \( 11,992.4 \).

Key Concepts

Understanding the Natural Logarithm

Logarithms are mathematical expressions that help us work with exponential relationships. The natural logarithm, denoted as \(\ln\), is a specific type of logarithm. It is based on the mathematical constant \(e\), which is approximately 2.71828.
When we say \(\ln(x)\), we're asking what power we need to raise \(e\) to in order to get \(x\). For example, if \(\ln(x) = y\), then \(e^y = x\).

• \(\ln(1) = 0\)
• \(\ln(e) = 1\)
• \(\ln(e^y) = y\)

Natural logarithms are frequently used in calculus and mathematical modeling because they simplify the process of dealing with growth and decay problems. In our exercise, we used a calculator to find \(\ln(28.1)\), which helps us scale and manipulate the function \(f(x)\).

Performing Function Evaluation

Function evaluation is a process where you substitute a given input into a function to find the output. In this exercise, our function \(f(x) = 75 + 3,570 \ln x\) requires evaluating at \(x = 28.1\).
To do this, we simply replace \(x\) with 28.1 in the function expression:
\(f(28.1) = 75 + 3,570 \ln(28.1)\).
This substitution lets us compute the output value by carrying out the operations according to the function's formula. Here’s a quick recap of how it works:

• Substitute the input value into the function, replacing \(x\) with the specific number.
• Calculate any inner functions, like \(\ln(x)\).
• Perform any multiplications or additions as dictated by the function.

This step-by-step approach helps ensure accuracy and makes complex operations more manageable.

Mastering Rounding Numbers

Rounding numbers is a fundamental skill in mathematics necessary for simplifying answers to a manageable level of precision. When rounding, you can choose to round up, down, or to the nearest value, typically depending on where the digit falls in a decimal.
In our solution, we rounded the final result \(11,992.4292\) to the nearest tenth. Here's how you can decide which direction to round:

• Identify the digit at the place value you are rounding to. Here it's the tenths place: the number 4 in 11,992.4292.
• Look at the digit immediately to the right of this place (the hundredths place). If this digit is 5 or more, round the tenths place up by one.
• If it's less than 5, leave the digit in the tenths place as it is.

In our case, the number in the hundredths place is 2, which means we keep 4 in the tenths place, resulting in \(11,992.4\). Rounding helps present data more clearly and avoids unnecessary complications with too many decimal places.