Problem 12 - Calculus with Concepts in Calculus

Question

Find the numerical value of the function at the given values of \(a\). $$ f(x)=\sqrt{\frac{192,000}{x}-6} ; a=10,000,21,729 $$

Step-by-Step Solution

Answer

f(10,000) \approx 3.633; f(21,729) \approx 1.684

Step 1: Identify the function

The function given is \( f(x) = \sqrt{\frac{192,000}{x} - 6} \). We need to evaluate this function for the given values of \( a \): 10,000 and 21,729.

Step 2: Substitute the first value

Substitute \( x = 10,000 \) into the function: \[ f(10,000) = \sqrt{\frac{192,000}{10,000} - 6} \]. Simplify the expression under the square root.

Step 3: Simplify the expression (for 10,000)

Calculate the fraction: \( \frac{192,000}{10,000} = 19.2 \). Substitute back into the expression: \[ f(10,000) = \sqrt{19.2 - 6} \]. Continue simplifying: \[ f(10,000) = \sqrt{13.2} \].

Step 4: Simplify using a calculator (for 10,000)

Use a calculator to find \( \sqrt{13.2} \). The value is approximately 3.633.

Step 5: Substitute the second value

Substitute \( x = 21,729 \) into the function: \[ f(21,729) = \sqrt{\frac{192,000}{21,729} - 6} \]. Simplify the expression under the square root.

Step 6: Simplify the expression (for 21,729)

Calculate the fraction: \( \frac{192,000}{21,729} \) is approximately 8.837. Substitute back into the expression: \[ f(21,729) = \sqrt{8.837 - 6} \]. Continue simplifying: \[ f(21,729) = \sqrt{2.837} \].

Step 7: Simplify using a calculator (for 21,729)

Use a calculator to find \( \sqrt{2.837} \). The value is approximately 1.684.

Key Concepts

Function Evaluation

At the heart of many mathematical problems lies the concept of function evaluation. When we are asked to evaluate a function, we are looking to find the output of the function for a specified input. In this exercise, we focus on the function \( f(x) = \sqrt{\frac{192,000}{x} - 6} \) and evaluate it for two different values of \(x\): 10,000 and 21,729.
Function evaluation involves substituting the given value into the function and then performing the necessary calculations. It is a fundamental concept that allows us to understand how functions behave and change depending on the input values. Once the values are substituted, computing the output becomes a matter of simplification and arithmetic.
Function evaluation is crucial in various fields like engineering, physics, and economics, where functions model real-world scenarios. Through function evaluation, we predict and analyze outcomes based on variable changes.

Square Root Function

The square root function is a very common function used to derive the principal square root of a non-negative number. This means it extracts the value that, when multiplied by itself, gives the original number. In our exercise, the square root component \( \sqrt{\frac{192,000}{x} - 6} \) plays a crucial role. We aim to determine the simplicity or complexity of the numerical result after the subtraction by 6 inside the square root.
Understanding the nature of square root functions is essential because they are both non-linear and have specific domains. For instance, the expression inside the square root must be non-negative to yield a real number, which imposes restrictions on potential values of \(x\). Thus, evaluating \( f(x) \) effectively means ensuring the expression \( \frac{192,000}{x} - 6 \) remains non-negative.
Seeing square roots in equations often indicates indicators like curves in graphical representations and can show up in an array of applications, from geometry to statistical dispersion measures.

Substitution Method

The substitution method is a straightforward strategy for solving numerical expressions by replacing variables with known values. This is exactly what we do in our problem when we input the values of \( a \), specifically 10,000 and 21,729, into the function.
To substitute means to replace every instance of the variable \(x\) with the value provided. This allows us to then proceed with arithmetic operations to simplify the function to an understandable and computable form. Once substitution has taken place, subsequent steps typically involve simplification processes to evaluate the resulting expressions easily.
Substitution is incredibly versatile and is applied in solving differential equations, integrals, and algebraic equations. It is an essential tool for anyone working through mathematical challenges, making complicated functions more manageable by breaking them down into simpler components.

Step-By-Step Simplification

Step-by-step simplification is a technique used to break down complex mathematical problems into manageable parts. Here, we follow a detailed sequence of calculations while evaluating the function \( f(x) = \sqrt{\frac{192,000}{x} - 6} \) for given values of \(a\).
Initially, we perform substitution to replace \(x\) with the given numbers like 10,000 or 21,729. Next, we compute arithmetic operations sequentially, first handling divisions before moving on to subtractions indicated inside the square root. This ensures the procedure remains clear and builds logically.

• Substitute the value of \(x\) into the function.
• Calculate the division \( \frac{192,000}{ \text{value} } \).
• Subtract 6 from the result.
• Take the square root of the outcome.

Ultimately, this method ensures transparency in mathematical reasoning and reduces the likelihood of errors. It is particularly beneficial in educational settings, providing learners with a clear roadmap of solving complex equations.