Sure! Let's solve the given literal equation [tex]\( t = 2 \pi \sqrt{\frac{m}{k}} \)[/tex] for [tex]\( m \)[/tex] step-by-step.
1. Starting with the given equation:
[tex]\[
t = 2 \pi \sqrt{\frac{m}{k}}
\][/tex]
2. Isolate the square root term:
To do this, we first divide both sides of the equation by [tex]\( 2\pi \)[/tex]:
[tex]\[
\frac{t}{2\pi} = \sqrt{\frac{m}{k}}
\][/tex]
3. Clear the square root:
Square both sides of the equation to eliminate the square root:
[tex]\[
\left( \frac{t}{2\pi} \right)^2 = \left(\sqrt{\frac{m}{k}}\right)^2
\][/tex]
[tex]\[
\left(\frac{t}{2\pi}\right)^2 = \frac{m}{k}
\][/tex]
4. Simplify the left side:
[tex]\[
\frac{t^2}{(2\pi)^2} = \frac{m}{k}
\][/tex]
5. Multiply through by [tex]\( k \)[/tex] to solve for [tex]\( m \)[/tex]:
[tex]\[
m = k \cdot \frac{t^2}{(2\pi)^2}
\][/tex]
6. Simplify the denominator:
[tex]\[
m = k \cdot \frac{t^2}{4\pi^2}
\][/tex]
So, the solution to the equation [tex]\( t = 2 \pi \sqrt{\frac{m}{k}} \)[/tex] for [tex]\( m \)[/tex] is:
[tex]\[
m = \frac{k t^2}{4 \pi^2}
\][/tex]
