Problem 94 - College Algebra with Modeling and Visualization

Question

Find the constant of proportionality \(k\) $$ y=k x^{3 / 2}, \text { and } y=96 \text { when } x=16 $$

Step-by-Step Solution

Answer

The constant of proportionality \( k \) is \( \frac{3}{2} \).

Step 1: Understand the Given Formula

We have the formula \( y = kx^{3/2} \). This suggests that \( y \) is proportional to \( x^{3/2} \) with \( k \) as the constant of proportionality that we need to find.

Step 2: Substitute Known Values

We are given that \( y = 96 \) and \( x = 16 \). Substitute these values into the equation: \[ 96 = k imes 16^{3/2} \]

Step 3: Calculate \( x^{3/2} \)

Calculate \( 16^{3/2} \). This involves two steps: first take the square root of 16, which is 4, and then cube the result: \( 4^3 = 64 \) Thus, \( 16^{3/2} = 64 \).

Step 4: Solve for \( k \)

Substitute \( 64 \) from the previous step into the equation: \[ 96 = k imes 64 \] Solve for \( k \) by dividing both sides by 64: \[ k = \frac{96}{64} = \frac{3}{2} \]

Step 5: Conclude the Calculation

Now we have found that the constant of proportionality \( k \) is \( \frac{3}{2} \).

Key Concepts

Algebraic Equations

Algebraic equations form the cornerstone of algebra, serving as the building blocks used to express relationships between variables. In our exercise, we are dealing with an equation where the relationship between two variables, \( y \) and \( x \), is defined by an equation: \[ y = kx^{3/2} \]This specific algebraic equation indicates that \( y \) is dependent on \( x \) raised to a certain power, with a proportionality constant \( k \) impacting the scale of this relationship. Recognizing the form of an equation is crucial since it allows us to apply suitable methods to solve it and find unknowns like \( k \).

• Firstly, identify the known and unknown variables.
• Formulate the equation and plug in known values.
• Use the equation structure to manipulate and solve for the unknown variable.

Algebraic equations may seem complex, but once dissected into smaller tasks, they become more manageable with the right approach.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself, such as \( x^n \), where \( x \) is the base and \( n \) is the exponent. Understanding exponents is crucial in this exercise, where the term \( x^{3/2} \) appears. To tackle this effectively:

• Break down the exponent: \( x^{3/2} \) means you take the square root of \( x \) and then raise that result to the third power.
• Practice sequential operations: for \( x = 16 \), calculate \( \sqrt{16} = 4 \), then \( 4^3 = 64 \).

Exponents can drastically change a base number's magnitude, and understanding how to work with them is important for solving equations that involve powers. It allows us to calculate values correctly and understand the relationship described by the equation.

Problem-Solving Technique

Effective problem-solving techniques can simplify solving algebraic equations like the one in our exercise. Here's a step-by-step strategy:1. **Understand the Problem**: Clearly define what you are asked to find. Here, we need to find the constant \( k \). 2. **Substitute Known Values**: Insert any known values into the equation to simplify. We utilized the given values of \( y = 96 \) and \( x = 16 \), making the problem tangible.3. **Execute Mathematical Operations**: - Calculate any complex terms involving exponents by breaking them into understandable chunks. - Complete arithmetic operations step-by-step to avoid mistakes.4. **Solve and Verify**: After computing, solve the equation for \( k \) and double-check your results to confirm accuracy.By systematically breaking down and tackling each part of the problem, you can confidently derive the correct solution. This technique helps ensure you don’t miss any steps, leading to accurate and reliable answers.