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Chapter 5: Problem 31

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube of \(x\). When \(x=3\) then \(y=1\). Find \(y\) when \(x=1\).

Short Answer

Expert verified
When \( x = 1 \), \( y = 27 \).

Step by step solution

01

Understand the Relationship

Since 'y varies inversely with the cube of x', this means the relationship can be expressed as \( y = \frac{k}{x^3} \), where \( k \) is the constant of variation.
02

Find the Constant of Variation

We are given that when \( x = 3 \), \( y = 1 \). Substitute these values into the equation to find \( k \): \( 1 = \frac{k}{3^3} \). This simplifies to \( 1 = \frac{k}{27} \). Solving for \( k \), we get \( k = 27 \).
03

Substitute and Solve for y

Now that we know \( k = 27 \), we substitute it back into the inverse variation equation: \( y = \frac{27}{x^3} \). We need to find \( y \) when \( x = 1 \). Substitute \( x = 1 \) into the equation: \( y = \frac{27}{1^3} = 27 \).
04

Verify the Solution

Verifying our steps, the calculated \( y \) when \( x = 1 \) should be consistent with the equation and initial conditions. Since \( y = \frac{27}{x^3} \) held true for \( (x=3, y=1) \), and consistent substitutions provided \( y = 27 \) for \( x = 1 \), our solution is verified correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant of Variation
The constant of variation plays a crucial role in understanding relationships between variables in inverse variation equations. The constant, usually denoted as \( k \), remains unchanged for a particular relationship. It helps in formulating how one variable affects another inversely. If you see a sentence like "\( y \) varies inversely with the cube of \( x \)," it implies: \( y = \frac{k}{x^3} \). Here:
  • \( y \) is inversely related to the cube of \( x \).
  • \( k \) remains constant across different values of \( x \) and \( y \).
Understanding the constant of variation is key to solving and verifying inverse variation problems. In this exercise, once \( k \) was found to be 27 for the given condition \((x=3, y=1)\), it applied uniformly when \( x \) changed to 1.
Cube of a Number
Cubing a number involves raising it to the power of three. It is calculated as \( x \times x \times x \), or \( x^3 \). Cubing:
  • Grows the value exponentially, making small numbers very small and larger ones much bigger.
  • Is essential in inverse relationships, especially when one variable inversely depends on another's cube.
For instance, in the problem, we calculated \( 3^3 \) to find that \( 27 \) influenced the value of \( y \). Remembering this concept helps in visualizing how alterations in \( x \) affect associated outcomes in inverse equations.
Algebraic Equations
Algebraic equations are mathematical statements that use symbols and variables to express relationships. In `inverse variation`, they allow us to set up formulas representing how variables interact:
  • The structure \( y = \frac{k}{x^3} \) is derived from understanding the relationship as \( y \) inversely varies with \( x^3 \).
  • Solving these equations typically involves isolating one variable by substituting known values and determining others.
In the given exercise, we first found \( k \) by substituting the initial conditions \((x=3, y=1)\), then solved for \( y \) using the determined \( k \) value. Mastery of such algebraic methods ensures accurate problem-solving across various mathematical challenges.

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Most popular questions from this chapter

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as the square of \(x\) and the square of \(z\) and when \(x=3\) and \(z=4,\) then \(y=72\).

For the following exercises, use a graph to help determine the domain of the functions. $$f(x)=\sqrt{\frac{(x+2)(x-3)}{x-1}}$$

For the following exercises, use the given information to find the unknown value. \(y\) varies jointly as the square of \(x\) and the cube of \(z\) and inversely as the square root of \(w\). When \(x=2\), \(z=2\), and \(w=64\), then \(y=12\). Find \(y\) when \(x=1\), \(z=3,\) and \(w=4\).

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x\) and \(z\) and inversely as \(w .\) When \(x=3\), \(z=5\), and \(w=6,\) then \(y=10\).

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: \(-1,1,3\) and \((2, f(2))=(2,4)\)
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