Let's begin by rewriting the given equation:
[tex]\[ y = \frac{a^2 b}{2} \][/tex]
We need to solve for [tex]\( a \)[/tex] in terms of [tex]\( y \)[/tex] and [tex]\( b \)[/tex]. Let's go through this step-by-step:
1. Multiply both sides of the equation by 2 to eliminate the denominator:
[tex]\[ 2y = a^2 b \][/tex]
2. Divide both sides by [tex]\( b \)[/tex] to isolate [tex]\( a^2 \)[/tex]:
[tex]\[ \frac{2y}{b} = a^2 \][/tex]
3. Take the square root of both sides to solve for [tex]\( a \)[/tex]:
[tex]\[ a = \pm \sqrt{\frac{2y}{b}} \][/tex]
Therefore, the solutions for [tex]\( a \)[/tex] are:
[tex]\[ a = \sqrt{\frac{2y}{b}} \quad \text{and} \quad a = -\sqrt{\frac{2y}{b}} \][/tex]
These are the two possible values for [tex]\( a \)[/tex] given the original equation [tex]\( y = \frac{a^2 b}{2} \)[/tex].
