Sure! Let's work through the problem step-by-step.
### 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 find the constant [tex]\( k \)[/tex], we'll use one of the given pairs of values from the table. Let’s use the values [tex]\( x = 1 \)[/tex] and [tex]\( y = 4 \)[/tex]:
[tex]\[ 4 = \frac{k}{1^2} \][/tex]
Simplifying this equation:
[tex]\[ k = 4 \][/tex]
So, the relationship between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] can be written as:
[tex]\[ y = \frac{4}{x^2} \][/tex]
This equation expresses [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
### Part (b)
Now, we need to find the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex].
Using the equation:
[tex]\[ y = \frac{4}{x^2} \][/tex]
we substitute [tex]\( y = 25 \)[/tex]:
[tex]\[ 25 = \frac{4}{x^2} \][/tex]
Rearranging to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ 25x^2 = 4 \][/tex]
[tex]\[ x^2 = \frac{4}{25} \][/tex]
Taking the positive square root of both sides:
[tex]\[ x = \sqrt{\frac{4}{25}} \][/tex]
[tex]\[ x = \frac{2}{5} \][/tex]
Therefore, the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex] is:
[tex]\[ x = \frac{2}{5} \][/tex]
