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For \[\text{Y}\] is inversely proportional to the square of \[\text{X}\] when \[\text{Y = 50, X = 2,}\] how do you find an equation connecting \[\text{Y}\] and \[\text{X}\]?

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Last updated date: 26th Feb 2024
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IVSAT 2024
Answer
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Hint: It is given as \[\text{A}\] is directly proportional to \[\text{B}\]. Then we can write this as \[\text{A }\!\!\alpha\!\!\text{ B}\] and we can equate it by introducing a constant between \[\text{A}\] and \[\text{B}\] as \[\text{A = KB}\] where \[\text{K}\] is constant.
For finding the value of \[\text{K}\] you need the values of \[\text{A}\] and \[\text{B}\] and then you can put it in equation to get the value of \[\text{K}\].

Complete step by step solution:It is given in the question that \[\text{Y}\] is inversely proportional to the square of \[\text{X}\] we can write it as \[\text{Y }\alpha \text{ }\dfrac{1}{{{x}^{2}}}\] and we can introduce constant \[\text{C}\]
\[\therefore \text{Y = }\dfrac{\text{C}}{{{x}^{2}}}\,......\,(1)\]
We have to find a equation connecting \[\text{Y}\] and \[\text{X}\] when \[\text{Y = 50}\] and \[\text{X = 2}\]
So, the proportional equation becomes
\[50\,=\,\dfrac{\text{C}}{{{\left( 2 \right)}^{2}}}\]
\[\text{C}\,\text{=}\,\text{50}\times {{2}^{2}}\]
\[\text{50}\times 4\]
\[\text{C}\,=\,200\]
Putting the value of \[\text{C}\] in equation \[(1)\] we get,
\[\text{Y}\,=\,\dfrac{200}{{{\text{X}}^{2}}}\], which could be written as \[{{x}^{2}}y\,\,=\,200\]
This is an equation connecting \[\text{Y}\] and \[\text{X}\] when \[\text{Y = 50}\] and \[\text{X = 2}\]

Additional Information:
When \[y\] is inversely proportional to the square of \[x\]. It means if \[x\] is increased two times then, the value of \[y\] decreases four times.
For example:
If \[x\,=\,2\]
\[y\,=\,\dfrac{\text{C}}{{{x}^{2}}}\,=\,y\,=\,\dfrac{\text{C}}{{{2}^{2}}}\,=\,\dfrac{\text{C}}{4}\]
The graph that represents this equation clearly.
seo images

Let us discuss the case where \[x\] is positive, if \[x\] is positive, then
As \[x\to \infty ,\,y\to 0\] and vice versa.
i.e if \[x\] gets larger, \[y\] gets smaller and vice versa.
Sometimes the question comes \[y\] is inversely proportional to \[x\] it can simply be written as \[y\,=\,\dfrac{\text{C}}{x}\,\]

Note:
When putting values of \[y\] and \[x\] in the given equation carefully solve and find the value of the constant you assumed.
It is not necessary to assume constant as \[\text{C}\] you can assume any variable you wish.
The sign \[\alpha \] is used for both inversely proportional and directly proportional questions.