In triangle LMN, we know the following:
- Angle M is a right angle (90 degrees).
- Angle L measures \( x \) degrees.
- The side \( LN \) (the hypotenuse) measures 22.5 units.
- The side \( NM \) (an opposite side to angle L) measures \( 3b \) units.
Given that \( \sin(x) = \frac{4}{5} \), we can use this information to set up the sine definition for angle L:
\[ \sin(L) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{NM}{LN} \]
Substituting the lengths we have:
\[ \sin(x) = \frac{3b}{22.5} \]
Now, plugging in the value of \( \sin(x) \):
\[ \frac{4}{5} = \frac{3b}{22.5} \]
To solve for \( b \), we can cross-multiply:
\[ 4 \cdot 22.5 = 3b \cdot 5 \] \[ 90 = 15b \]
Now, isolate \( b \):
\[ b = \frac{90}{15} = 6 \]
Thus, the value of \( b \) is:
\[ \boxed{6} \]
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