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Chapter 1: Problem 43

Finding a Mathematical Model In Exercises \(41-50\) , find a mathematical model for the verbal statement. \(y\) varies inversely as the square of \(x\)

Short Answer

Expert verified
The mathematical model for the statement \(y\) varies inversely as the square of \(x\) is \(yx^2=k\).

Step by step solution

01

Understand inverse variation

The first step is to understand that 'varies inversely' means that as one quantity increases, the other decreases proportionately. So for some constant \(k\), the product of the two quantities is always equal to \(k\). More specifically, here we are dealing with 'varies inversely as the square', which means \(y\) times the square of \(x\) (denoted as \(x^2\)) equals \(k\) for some constant \(k\).
02

Write the equation

Given that \(y\) varies inversely as the square of \(x\), we can write this relationship as \(yx^2=k\). This is a mathematical model for the verbal statement.
03

Final expression

The final mathematical model which expresses the statement '\(y\) varies inversely as the square of \(x\)' is the equation \(yx^2=k\). This model allows to calculate the value of \(y\) for any given value of \(x\) and a set constant \(k\), or vice versa.

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Key Concepts

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

Square of a Number
In mathematics, when we refer to the "square of a number", we are talking about multiplying the number by itself. For example, the square of a number, let's say 3, is represented as \(3^2\), which equals 9. Squares are an essential part of various mathematical concepts and operations.
They find applications in geometry, algebra, and even in real-world scenarios.
  • The notation \(x^2\) is used to represent the square of \(x\).
  • Squaring a negative number results in a positive number. For instance, \((-2)^2 = 4\).
  • In geometry, the area of a square (a four-sided figure with equal sides) can be calculated by squaring the length of one of its sides.
Understanding squares is crucial when dealing with powers, polynomials, and specifically in this case, inverse variation problems involving the square of a number.
Mathematical Modeling
Mathematical modeling is the process of using mathematical equations and expressions to describe real-world phenomena or verbal statements. In the context of this exercise, we take a verbal statement, "y varies inversely as the square of x," and convert it into a mathematical model.
Here's a straightforward way to understand this conversion:
  • An inverse variation means one variable increases while the other decreases, maintaining an inverse relationship.
  • The expression \(y \propto \frac{1}{x^2}\) shows that \(y\) is inversely proportional to the square of \(x\).
  • To express this model using an equation, we equate the product of \(y\) and \(x^2\) with a constant \(k\), leading to \(yx^2 = k\).
By establishing the model \(yx^2 = k\), you can easily solve for any unknown in the relationship given the other values and the constant. This concept is fundamental in science, engineering, and economics for predicting and understanding system behaviors.
Constant of Variation
The constant of variation, often denoted as \(k\), is a fixed value that defines the relationship between variables in a variation equation. This concept is significant because it remains consistent regardless of the values of the variables, given that the relationship holds true.
In the equation \(yx^2 = k\):
  • \(k\) represents the constant of variation that links \(y\) and \(x^2\).
  • For any set of \(y\) and \(x\), the equation will always result in the same \(k\) as long as \(y\) varies inversely as \(x^2\).
  • Knowing \(k\) allows us to calculate one variable if the others are known. For instance, if \(k\) and \(x\) are given, we can find \(y\).
Understanding the constant of variation helps in verifying and applying the mathematical model effectively. It allows for accurate predictions and insight into how changing one variable impacts another in inverse variation scenarios.

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Most popular questions from this chapter

Describing Profits Management originally predicted that the profits from the sales of a new product would be approximated by the graph of the function \(f\) shown. The actual profits are shown by the function \(g\) along with a verbal description. Use the concepts of transformations of graphs to write \(g\) in terms of \(f .\) (a) The profits were only three-fourths as large as expected. (b) The profits were consistently \(\$ 10,000\) greater than predicted. (c) There was a two-year delay in the introduction of the product. After sales began, profits grew as expected.

Evaluating a Difference Quotient In Exerrises \(77-84\) , find the difference quotient and simplify your answer. $$f(x)=x^{3}+3 x, \quad \frac{f(x+h)-f(x)}{h}, \quad h \neq 0$$

Cost, Revenue, and Profit A roofing contractor purchases a shingle delivery truck with a shingle elevator for \(\$ 42,000\) . The vehicle requires an average expenditure of \(\$ 9.50\) per hour for fuel and maintenance, and the operator is paid \(\$ 11.50\) per hour. (a) Write a linear equation giving the total cost \(C\) of operating this equipment for \(t\) hours. (Include the purchase cost of the equipment.) (b) Assuming that customers are charged \(\$ 45\) per hour of machine use, write an equation for the revenue \(R\) derived from \(t\) hours of use. (c) Use the formula for profit \(P=R-C\) to write an equation for the profit derived from \(t\) hours of use. (d) Use the result of part (c) to find the break-even point- -that is, the number of hours this equipment must be used to yield a profit of 0 dollars.

Finding an Equation of a Line ,find an equation of the line passing through the points. Sketch the line. $$(2,-1),\left(\frac{1}{3},-1\right)$$

Evaluating a Difference Quotient In Exerciseses \(77-84\) , find the difference quotient and simplify your answer. $$g(x)=\frac{1}{x^{2}}, \quad \frac{g(x)-g(3)}{x-3}, \quad x \neq 3$$
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