To find the level curve of the function $f(x, y) = x^2 + y^2 + 37.3$ that passes through the point $(1, 1)$, we first calculate the value of the function at that point:

$f(1, 1) = 1^2 + 1^2 + 37.3 = 1 + 1 + 37.3 = 39.3$

The equation of the level curve is given by setting the function equal to this value:

$x^2 + y^2 + 37.3 = 39.3$

Simplifying, we get:

$x^2 + y^2 = 2$

So, the equation of the level curve is $x^2 + y^2 = 2$.

Do you want more details on level curves or related concepts? Here are some questions for further exploration:

1. What are level curves, and how do they relate to functions of multiple variables?
2. How can you visualize the level curve of this function in a graph?
3. What other points can be used to determine different level curves for this function?
4. How does changing the constant in the function affect the level curve?
5. Can you explain how to find level curves for a function that is not in standard form?

Tip: When analyzing level curves, remember they represent sets of points where the function takes on a constant value, which can help in understanding the function's behavior.