Question

Suppose that $y$ varies inversely with the square root of $x$ , and $y = 50$ when $x = 4$. How do you find $y$ when $x = 5$?

Answer 1

Hint: Inverse variation problems are solved using the equation \[y = \dfrac{k}{x}\].
When dealing with word problems, you should consider using variables other than x and y, you should use variables that are relevant to the problem being solved.
Also read the problem carefully to determine if there are any other changes in the inverse variation equation, such as squares, cubes, or square roots.
Use the information given in the problem to find the value of k, called the constant of variation or the constant of proportionality.
Rewrite the equation by substituting in the value of k found. Use the equation found and the remaining information given in the problem to answer the question asked.

Complete step-by-step solution:
When two quantities vary inversely, their products are always equal to a constant, which we can call $k$.
If the square root of $x$ and the $y$ vary inversely, this means that the product of the square root of $x$ and $y$ will equal to $k$. Now we can write the equation for inverse variation
$ \Rightarrow \sqrt {x\,} y = k.............\left( 1 \right)$
We are told that when $y = 50\,x = 4$ .
Now we can substitute these values in $\left( 1 \right)$ equation we get
$ \Rightarrow \sqrt 4 \times 50 = k$
We know the value of $\sqrt 4 $ is $2$
Therefore, $ \Rightarrow 2 \times 50 = k$
On multiplying we get
$ \Rightarrow k = 100$
Now we have to find the Value of $y$ when $x = 5$
By using the first equation we get
$ \Rightarrow \sqrt 5 \, \times y = 100$
Now on solving the square root without using calculator
First bring the term $y$ to the RHS, we get
$ \Rightarrow \sqrt 5 = \dfrac{{100}}{y}$
Now squaring on both sides we get
$ \Rightarrow {\left( {\sqrt 5 } \right)^2} = \dfrac{{{{\left( {100} \right)}^2}}}{{{y^2}}}$
We can write ${\left( {100} \right)^2}$ as ${\left( {10} \right)^4}$ and also we can cancel the square and square root
$ \Rightarrow 5 = \dfrac{{{{10}^4}}}{{{y^2}}}$
Now from the above equation we can find the value of $y$
$ \Rightarrow {y^2} = \dfrac{{{{10}^4}}}{5}$
On dividing them we get
$ \Rightarrow {y^2} = 2000$
Now taking square root on both sides we get
$ \Rightarrow y = \sqrt {2000} $
This can be written as
$ \Rightarrow y = 10\sqrt {20} $

Hence the value of $y$ is $10\sqrt {20} $.

Note: We say that $y$ varies inversely with $x$ if $y$ is expressed as the product of some constant number$k$ and the reciprocal of $x$ .However, the value of $k$ can’t be equal zero, i.e $k \ne 0$.
Isolating $k$ on one side, it becomes clear that $k$ is the fixed product of $x$ and $y$ . That means, multiplying $x$ and $y$ always yields a constant output of $k$.
Also The value of $y$ varies inversely to $\sqrt x $ which denotes the positive square root which also implies that the value of $y$ is positive.