Question

If \[y\] varies inversely as \[x\], \[y = 12\] as \[x = 2\]. How do you find \[y\] for the \[x\]-value of \[6\]?

Answer 1

Hint:Here, we will use the given information to form a relation between the two variables such that the relationship will be in the form of an unknown constant. Then we will substitute the given points in that equation to get the value of the unknown constant. Finally, substituting the given value of \[x\], we will obtain the required value of \[y\] which will be the final answer.

Complete step-by-step solution:
According to the question, \[y\] varies inversely as \[x\]. Writing this statement mathematically, we have
\[y \propto \dfrac{1}{x}\]
Replacing the proportionality sign with some unknown constant \[c\], we get
\[y = \dfrac{c}{x}\]………………………..\[\left( 1 \right)\]
In the above question, we have been given that \[y = 12\] as \[x = 2\].
Therefore, substituting \[y = 12\] and \[x = 2\] in the above equation, we get
\[ \Rightarrow 12 = \dfrac{c}{2}\]
Multiplying 2 on both sides, we get
\[ \Rightarrow c = 12 \times 2\]
\[ \Rightarrow c = 24\]
Putting this value in equation \[\left( 1 \right)\], we get
\[y = \dfrac{{24}}{x}\]
Now, according to the question, we have to find the value of \[y\] for the \[x\]-value of \[6\].
Therefore, substituting \[x = 6\] in the above equation, we get
\[ \Rightarrow y = \dfrac{{24}}{6}\]
Dividing 24 by 6, we get
\[ \Rightarrow y = 4\]

Hence, the value of \[y\] for the \[x\]-value of \[6\] is equal to \[4\].

Additional information:
The graph between two quantities, which are inversely proportional to each other, is a rectangular hyperbola. It is as shown in the below diagram

Note: The inverse proportionality is observed for many physical phenomena occurring in day-to-day life. For example, in a team aimed at completing a specific task, the time required for completing the task is inversely proportional to the number of members working in the team. Also, the time required by a moving vehicle to cover a particular distance is inversely proportional to the speed of the vehicle.