To evaluate the expression \( y^2 + \frac{55}{x} - 12 \) at \( x = 11 \) and \( y = 5 \), we can substitute these values into the expression.
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Substitute \( y = 5 \): \[ y^2 = 5^2 = 25 \]
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Substitute \( x = 11 \): \[ \frac{55}{x} = \frac{55}{11} = 5 \]
Now, we can substitute both values back into the expression: \[ y^2 + \frac{55}{x} - 12 = 25 + 5 - 12 \]
Now, calculate: \[ 25 + 5 = 30 \] \[ 30 - 12 = 18 \]
Thus, the value of the expression when \( x = 11 \) and \( y = 5 \) is \( \boxed{18} \).
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