[tex]\[ y \][/tex] is inversely proportional to the square of [tex]\[ x \][/tex].

It is known that [tex]\[ y = 12 \][/tex] for a particular value of [tex]\[ x \][/tex].

When [tex]\[ x \][/tex] is increased by [tex]\[ 100\% \][/tex], find:

(i) the value of [tex]\[ y \][/tex]



Answer :

To solve this problem, let's follow the detailed steps one by one.

1. Understand Inverse Proportionality:
Given that [tex]\( y \)[/tex] is inversely proportional to the square of [tex]\( x \)[/tex], we can write this relationship as:
[tex]\[ y = \frac{k}{x^2} \][/tex]
where [tex]\( k \)[/tex] is a constant.

2. Determine the Constant [tex]\( k \)[/tex]:
We are given that [tex]\( y = 12 \)[/tex] for a particular value of [tex]\( x \)[/tex]. Let's denote this initial value of [tex]\( x \)[/tex] as [tex]\( x_{\text{initial}} \)[/tex].
[tex]\[ y_{\text{initial}} = 12 \][/tex]
Assuming [tex]\( x_{\text{initial}} = 1 \)[/tex] for simplicity (based on the relationship given),
[tex]\[ 12 = \frac{k}{1^2} \][/tex]
Solving for [tex]\( k \)[/tex],
[tex]\[ k = 12 \][/tex]

3. Increase [tex]\( x \)[/tex] by 100%:
Increasing [tex]\( x \)[/tex] by 100% means that the new value of [tex]\( x \)[/tex] is double the initial value. So, let's denote the new value of [tex]\( x \)[/tex] as [tex]\( x_{\text{new}} \)[/tex].
[tex]\[ x_{\text{new}} = 2 \cdot x_{\text{initial}} = 2 \cdot 1 = 2 \][/tex]

4. Find the New Value of [tex]\( y \)[/tex]:
We need to determine the new value of [tex]\( y \)[/tex] when [tex]\( x_{\text{new}} = 2 \)[/tex]. Using the previously found constant [tex]\( k = 12 \)[/tex],
[tex]\[ y_{\text{new}} = \frac{k}{(x_{\text{new}})^2} = \frac{12}{2^2} = \frac{12}{4} = 3.0 \][/tex]

So, the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is increased by 100% is:
[tex]\[ y = 3.0 \][/tex]