The cube of [tex]$m$[/tex] varies inversely as the square root of [tex]$n$[/tex]. Which two equations model this relationship?
[tex]$m^3=\frac{k}{\sqrt{n}}$[/tex]
[tex]$\frac{m^2}{\sqrt{n}}=k$[/tex]
[tex]$m=\frac{k}{n}$[/tex]
[tex]$m^3=k n^{\frac{1}{3}}$[/tex]
[tex]$m^3 n^{\frac{1}{2}}=k$[/tex]
[tex]$m n=k$[/tex]
Answer (2)
The cube of m varies inversely as the square root of n , which means m 3 is inversely proportional to n .
Write the general equation for inverse variation: m 3 = n k .
Rewrite the equation by multiplying both sides by n : m 3 n = k .
Express n as n 2 1 to get the second equation: m 3 n 2 1 = k .
The two equations that model the relationship are: m 3 = n k and m 3 n 2 1 = k .
Explanation
Understanding the Problem We are given that the cube of m varies inversely as the square root of n . This means that m 3 is inversely proportional to n . We need to find two equations that model this relationship.
Writing the General Equation The general form of an inverse variation is y = x k , where k is the constant of variation. In this case, m 3 varies inversely as n , so we can write the equation as: m 3 = n k .
Rewriting the Equation To find the second equation, we can multiply both sides of the equation m 3 = n k by n : m 3 × n = k
Using Exponent Notation We can also rewrite n as n 2 1 , so the equation becomes: m 3 n 2 1 = k
Identifying the Equations Therefore, the two equations that model the relationship are m 3 = n k and m 3 n 2 1 = k .
Examples
Imagine you are designing a water pump system. The power ( m 3 ) needed to run the pump varies inversely with the square root of the water source's purity level ( n ). If you know the power required for a certain purity level, you can use these equations to predict the power needed for different purity levels, helping you choose the right pump and optimize energy usage. This concept applies to various engineering and environmental scenarios where inverse relationships are crucial for efficient design and resource management.
The two equations modeling the relationship are m 3 = n k and m 3 n 2 1 = k .
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