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Chapter 1: Problem 23

Find a direct variation model that relates \(y\) and \(x\) $$x=4, y=8 \pi$$

Short Answer

Expert verified
The direct variation model that relates \(y\) and \(x\) is \(y = 2 \pi x\).

Step by step solution

01

Understand the Direct Variation

The equation of the direct variation model is as follows: \(y=kx\), where \(k\) is the constant of variation. We can find \(k\) by rearranging the equation to \(k=y/x\) when \(x鈮�0\).
02

Substitute and Solve

Plugging the given values into the equation, we have: \(k = 8 \pi / 4 = 2 \pi\).
03

Write the Final Variation Model

Substituting \(k= 2 \pi\) back into the direct variation model equation \(y=kx\), the final variation model that relates \(y\) and \(x\) is: \(y=2\pi x\).

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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, often denoted as \(k\), is a crucial part of understanding direct variations. In a direct variation, the concept states that one variable changes directly as another does.
This means if you multiply \(x\) by a certain number, \(y\) will also increase or decrease by this constant.
In the equation \(y = kx\), \(k\) represents this fixed ratio between \(y\) and \(x\).

  • When \(x\) is not zero, we can find \(k\) by using the formula \(k = \frac{y}{x}\).
  • This ratio remains the same for all values of \(x\) and \(y\) following the model \(y = kx\).
  • If \(k\) is positive, \(y\) increases as \(x\) increases. Similarly, if \(k\) is negative, \(y\) decreases as \(x\) increases.
In our exercise, substituting the given values of \(y = 8\pi\) and \(x = 4\), we calculate \(k = \frac{8\pi}{4} = 2\pi\), indicating a proportional relationship where for every unit increase in \(x\), \(y\) increases by \(2\pi\).
Variation Model
A variation model is a mathematical representation that shows how two variables are related to each other. Direct variation models are among the simplest.
The formula \(y = kx\) gives you a clear representation of how one variable (\(y\)) is affected by changes in another (\(x\)).

  • These models are linear, meaning if you plotted them on a graph, they would form a straight line through the origin (0, 0).
  • The slope of this line is represented by the constant of variation, \(k\).
  • Using a variation model helps in predicting the value of \(y\) for any given \(x\).
In the case of the problem provided, once \(k\) was found to be \(2\pi\), the variation model becomes \(y = 2\pi x\). This tells you exactly how \(y\) behaves as \(x\) changes. It's a precise and straightforward way of representing and understanding the relationship.
Mathematical Modeling
Mathematical modeling involves creating equations that represent real-world situations. It allows us to predict, understand, and communicate the behavior of different systems.
When a scenario involves direct variation, like our example with \(x = 4\) and \(y = 8\pi\), a direct variation model is an ideal choice.

  • Mathematical models like these simplify complex relationships into understandable forms.
  • They are used widely in various fields such as physics, economics, engineering, and more.
  • By transforming abstract ideas into tangible equations, we can analyze and interpret data with greater ease.
In the exercise, mathematical modeling helped to establish a clear equation \(y = 2\pi x\), encapsulating the relationship between \(x\) and \(y\). This model can now be used to predict \(y\) given any \(x\), making the abstract concept tangible and actionable.

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Most popular questions from this chapter

Beam Load The maximum load that can be safely supported by a horizontal beam varies jointly as the width of the beam and the square of its depth and inversely as the length of the beam. Determine the changes in the maximum safe load under the following conditions. A. The width and length of the beam are doubled. B. The width and depth of the beam are doubled.

Consider \(f(x)=\sqrt{x-1}\) and \(g(x)=\frac{1}{\sqrt{x-1}}\) Why are the domains of \(f\) and \(g\) different?

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(z\) varies jointly as \(x\) and \(y .(z=64 \text { when } x=4\) and \(y=8 .)\)

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped.

A rectangle is bounded by the \(x\) -axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x,\) and graphically determine the domain of the function.
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