Certainly! Let's solve the given problem in a detailed, step-by-step manner.
### Part a) Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
1. Understanding the relationship:
Since [tex]\( y \)[/tex] is inversely proportional to the square of [tex]\( x \)[/tex], we can write:
[tex]\[
y = \frac{k}{x^2}
\][/tex]
where [tex]\( k \)[/tex] is a constant.
2. Finding the constant [tex]\( k \)[/tex]:
We can use any of the given values in the table to find [tex]\( k \)[/tex]. Let's use the first value from the table where [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]
3. Solve for [tex]\( k \)[/tex]:
[tex]\[
k = 4
\][/tex]
4. Formulate [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
Substituting [tex]\( k \)[/tex] back into the equation gives:
[tex]\[
y = \frac{4}{x^2}
\][/tex]
Thus, we have expressed [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = \frac{4}{x^2}
\][/tex]
### Part b) Work out the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex].
1. Set up the equation:
Using the equation derived in part (a), substitute [tex]\( y = 25 \)[/tex]:
[tex]\[
25 = \frac{4}{x^2}
\][/tex]
2. Solve for [tex]\( x^2 \)[/tex]:
Rearrange the equation to solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{4}{25}
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], take the positive square root of both sides:
[tex]\[
x = \sqrt{\frac{4}{25}}
\][/tex]
Simplify under the square root:
[tex]\[
x = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5} = 0.4
\][/tex]
Therefore, the positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex] is:
[tex]\[
x = 0.4
\][/tex]
To summarize:
a) [tex]\( y = \frac{4}{x^2} \)[/tex]
b) The positive value of [tex]\( x \)[/tex] when [tex]\( y = 25 \)[/tex] is [tex]\( x = 0.4 \)[/tex].
