Answer :
Let's analyze the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and find the necessary values step-by-step.
### Part a: Find an equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
Given that [tex]\( y \)[/tex] is inversely proportional to the square of [tex]\( x \)[/tex], we can express this relationship mathematically as follows:
[tex]\[ y = \frac{k}{x^2} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
We can determine the value of [tex]\( k \)[/tex] using any given pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values from the table. Let's use the first pair [tex]\((1, 16)\)[/tex]:
When [tex]\( x = 1 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[ 16 = \frac{k}{1^2} \][/tex]
[tex]\[ k = 16 \][/tex]
Now that we have determined [tex]\( k = 16 \)[/tex], we can write the equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = \frac{16}{x^2} \][/tex]
### Part b: Find the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex].
We need to find [tex]\( x \)[/tex] such that when [tex]\( y = 25 \)[/tex], the equation [tex]\( y = \frac{16}{x^2} \)[/tex] holds true. Plug [tex]\( y = 25 \)[/tex] into the equation and solve for [tex]\( x \)[/tex]:
Starting with the equation:
[tex]\[ 25 = \frac{16}{x^2} \][/tex]
Rearranging to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ 25 x^2 = 16 \][/tex]
[tex]\[ x^2 = \frac{16}{25} \][/tex]
Taking the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{\frac{16}{25}} \][/tex]
[tex]\[ x = \frac{\sqrt{16}}{\sqrt{25}} \][/tex]
[tex]\[ x = \frac{4}{5} \][/tex]
[tex]\[ x = 0.8 \][/tex]
Thus, the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex] is:
[tex]\[ x = 0.8 \][/tex]
### Part a: Find an equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
Given that [tex]\( y \)[/tex] is inversely proportional to the square of [tex]\( x \)[/tex], we can express this relationship mathematically as follows:
[tex]\[ y = \frac{k}{x^2} \][/tex]
where [tex]\( k \)[/tex] is a constant of proportionality.
We can determine the value of [tex]\( k \)[/tex] using any given pair of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values from the table. Let's use the first pair [tex]\((1, 16)\)[/tex]:
When [tex]\( x = 1 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[ 16 = \frac{k}{1^2} \][/tex]
[tex]\[ k = 16 \][/tex]
Now that we have determined [tex]\( k = 16 \)[/tex], we can write the equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = \frac{16}{x^2} \][/tex]
### Part b: Find the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex].
We need to find [tex]\( x \)[/tex] such that when [tex]\( y = 25 \)[/tex], the equation [tex]\( y = \frac{16}{x^2} \)[/tex] holds true. Plug [tex]\( y = 25 \)[/tex] into the equation and solve for [tex]\( x \)[/tex]:
Starting with the equation:
[tex]\[ 25 = \frac{16}{x^2} \][/tex]
Rearranging to solve for [tex]\( x^2 \)[/tex]:
[tex]\[ 25 x^2 = 16 \][/tex]
[tex]\[ x^2 = \frac{16}{25} \][/tex]
Taking the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{\frac{16}{25}} \][/tex]
[tex]\[ x = \frac{\sqrt{16}}{\sqrt{25}} \][/tex]
[tex]\[ x = \frac{4}{5} \][/tex]
[tex]\[ x = 0.8 \][/tex]
Thus, the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex] is:
[tex]\[ x = 0.8 \][/tex]
