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Chapter 3: Problem 30

Express each direct variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 1. (OBJECTIVE 1) \(A\) varies directly with \(z\). If \(A=30\) when \(z=5,\) find \(A\) when \(z=9\).

Short Answer

Expert verified
The value of \(A\) when \(z = 9\) is \(54\).

Step by step solution

01

Formulate the direct variation equation

Direct variation can be expressed in the equation as \(A = kz\). Where \(k\) is the constant or the coefficient of variation.
02

Calculate the constant of variation \(k\)

Given that \(A = 30\) when \(z = 5\), the constant of variation \(k\) can be determined by dividing \(A\) by \(z\) i.e., \(k = A/z = 30/5 = 6\). So, the equation becomes \(A = 6z\).
03

Calculate the new value of \(A\) when \(z = 9\)

Substitute \(z = 9\) into the equation \(A = 6z\), the value of \(A\) comes out as \(A = 6*9 = 54\).

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Key Concepts

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

Variation Equation
In direct variation, one variable changes directly as another variable changes. The relationship between these two variables can be represented through a variation equation. This equation follows a simple linear format and is derived from experimenting or given data.

To express a direct variation as a mathematical equation, we use the formula:
  • Let the first variable, often represented as \(A\), vary directly with another variable, say \(z\). This gives us the equation \(A = kz\).
  • Here, \(k\) is the constant of variation, representing how much \(A\) changes as \(z\) changes.
Understanding how to set up this equation forms the foundation for solving direct variation problems effectively.
Constant of Variation
The Constant of Variation, \(k\), is like the "multiplier" that tells us how one variable changes in relation to another. Once we know \(k\), it becomes the key to unlocking the relationship between the two variables.

Here's how to find the Constant of Variation:
  • Based on the given equation \(A = kz\), if you know specific values of \(A\) and \(z\), you can calculate \(k\) by rearranging the equation: \(k = \frac{A}{z}\).
  • For our example, when \(A = 30\) and \(z = 5\), this leads to \(k = \frac{30}{5} = 6\).
Knowing \(k\) allows you to predict or calculate new values of \(A\) when \(z\) changes.
Mathematical Formulation
Mathematical Formulation refers to the specific creation of equations based on given relationships or data. When given a direct variation situation, transforming words and numbers into an equation is an essential step.

Here's the process:
  • Identify that you have a direct variation problem, which always involves a "varies directly" relation.
  • Translate this into the equation \(A = kz\), ensuring proper identification of variables and constants.
  • Using initial data such as \(A = 30\) and \(z = 5\), solve for \(k\) to completely form the equation: \(A = 6z\). This is your mathematical formulation.
This process of creating a formula is crucial for problem-solving efficiency and clarity.
Variable Relationship
The relationship between variables in direct variation is straightforward and proportional. It means that if one variable doubles, the other also doubles, maintaining a consistent ratio due to the Constant of Variation.

The characteristics of Variable Relationships in direct variation include:
  • A linear relationship represented by the equation: unchanging slope describes how the increase or decrease in one variable affects the other.
  • Dependence of one variable on another is clearly defined, as seen in \(A = 6z\), where any value of \(z\) helps predict the value of \(A\).
  • Application to new scenarios: For example, if \(z\) becomes 9, simply plug this value into the equation to find \(A\): \(A = 6 \times 9 = 54\).
Understanding this relationship empowers you to solve problems involving direct variation confidently.

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Most popular questions from this chapter

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(y\) varies directly with \(x^{3} .\) If \(y=16\) when \(x=2,\) find \(y\) when \(x=3\).

Express each combined variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 4. (OBJECTIVE 4) \(p\) varies directly with \(q\) and inversely with \(r .\) If \(p=5\) when \(q=1\) and \(r=6,\) find \(p\) when \(q=5\) and \(r=10\)

Research A psychology major found that the time \(t\) in seconds that it took a white rat to complete a maze was related to the number of trials \(n\) the rat had been given by the equation \(t=25-0.25 n\) a. Complete the table of values and then graph the equation. b. Complete this sentence: From the graph, we see that the more trials the rat had, the. c. From the graph, estimate the time it will take the rat to complete the maze on its 32nd trial. d. Interpret the meaning of the \(y\)-intercept.

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$P(2,-7) \text { and } Q(-3,12)$$

Explain the process used to find the \(x\)- and \(y\)-intercepts of the graph of a line.
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