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]
