To find the solutions for the equation [tex]\(\sqrt{\left(\frac{400}{\omega-2}\right)\left(\frac{400}{L}+2\right)} = 20\)[/tex], let's solve this step-by-step.
First, simplify the equation by removing the square root. Square both sides to get:
[tex]\[ \left(\frac{400}{\omega-2}\right)\left(\frac{400}{L}+2\right) = 400 \][/tex]
Next, distribute the left side:
[tex]\[ \frac{400 \cdot \left(\frac{400}{L} + 2\right)}{\omega - 2} = 400 \][/tex]
[tex]\[ \frac{400 \cdot \frac{400}{L} + 800}{\omega - 2}= 400 \][/tex]
[tex]\[ \frac{160000 + 800L}{L(\omega - 2)} = 400 \][/tex]
Now multiply both sides by [tex]\(L(\omega - 2)\)[/tex] to eliminate the fraction:
[tex]\[ 160000 + 800L = 400L(\omega - 2) \][/tex]
Distribute on the right side:
[tex]\[ 160000 + 800L = 400L\omega - 800L \][/tex]
Combine terms with [tex]\(L\)[/tex] on the right side:
[tex]\[ 160000 + 1600L = 400L\omega \][/tex]
Solve for [tex]\(\omega\)[/tex]:
[tex]\[ \omega = \frac{160000 + 1600L}{400L} \][/tex]
[tex]\[ \omega = \frac{160000}{400L} + \frac{1600L}{400L} \][/tex]
[tex]\[ \omega = \frac{160000}{400L} + 4 \][/tex]
[tex]\[ \omega = \frac{400}{L} + 4 \][/tex]
Therefore, the solutions for the given equation are:
[tex]\[ \omega = 4 + \frac{400}{L} \][/tex]
and [tex]\(L\)[/tex] is any real number such that the expressions are defined. Hence, the solution set is:
[tex]\[ \left(4 + \frac{400}{L}, L\right) \][/tex]
