To make [tex]\(x\)[/tex] the subject of the formula [tex]\(S = w \sqrt{a^2 - x^2}\)[/tex], we need to isolate [tex]\(x\)[/tex] on one side of the equation. Here are the steps to achieve this:
1. Starting equation:
[tex]\[
S = w \sqrt{a^2 - x^2}
\][/tex]
2. Isolate the square root term: Divide both sides by [tex]\(w\)[/tex]:
[tex]\[
\frac{S}{w} = \sqrt{a^2 - x^2}
\][/tex]
3. Eliminate the square root: Square both sides to get rid of the square root:
[tex]\[
\left(\frac{S}{w}\right)^2 = a^2 - x^2
\][/tex]
This simplifies to:
[tex]\[
\frac{S^2}{w^2} = a^2 - x^2
\][/tex]
4. Rearrange to isolate [tex]\(x^2\)[/tex]: Move [tex]\(x^2\)[/tex] to one side and [tex]\(\frac{S^2}{w^2}\)[/tex] to the other side:
[tex]\[
x^2 = a^2 - \frac{S^2}{w^2}
\][/tex]
5. Solve for [tex]\(x\)[/tex]: Take the square root of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \pm \sqrt{a^2 - \frac{S^2}{w^2}}
\][/tex]
Thus, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = \sqrt{a^2 - \frac{S^2}{w^2}} \quad \text{and} \quad x = -\sqrt{a^2 - \frac{S^2}{w^2}}
\][/tex]
So, the values of [tex]\(x\)[/tex] in terms of [tex]\(S\)[/tex], [tex]\(w\)[/tex], and [tex]\(a\)[/tex] are:
[tex]\[
x = \pm \sqrt{a^2 - \frac{S^2}{w^2}}
\][/tex]
or, more explicitly,
[tex]\[
x = \sqrt{a^2 - \frac{S^2}{w^2}} \quad \text{or} \quad x = -\sqrt{a^2 - \frac{S^2}{w^2}}
\][/tex]
These are the two possible solutions for [tex]\(x\)[/tex].
