To solve this problem, we will follow the steps to address both parts (a) and (b) using the given relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
### Part (a)
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.
To determine the constant [tex]\( k \)[/tex], we use one set of values from the table. Let's use [tex]\( x = 1 \)[/tex] and [tex]\( y = 4 \)[/tex].
Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = 4 \)[/tex] into the equation:
[tex]\[
4 = \frac{k}{1^2}
\][/tex]
[tex]\[
4 = k
\][/tex]
Thus, the constant [tex]\( k = 4 \)[/tex].
Therefore, the equation expressing [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] is:
[tex]\[
y = \frac{4}{x^2}
\][/tex]
### Part (b)
To find the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex], we use the relationship established:
[tex]\[
y = \frac{4}{x^2}
\][/tex]
Substitute [tex]\( y = 25 \)[/tex] into the equation:
[tex]\[
25 = \frac{4}{x^2}
\][/tex]
To find [tex]\( x^2 \)[/tex], multiply both sides of the equation by [tex]\( x^2 \)[/tex]:
[tex]\[
25x^2 = 4
\][/tex]
Now, solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{4}{25}
\][/tex]
Taking the square root of both sides to find [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{\frac{4}{25}}
\][/tex]
[tex]\[
x = \frac{2}{5}
\][/tex]
[tex]\[
x = 0.4
\][/tex]
Thus, the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex] is [tex]\( x = 0.4 \)[/tex].
