Find the domain of the following function: \[ z=\sqrt{400-16 x^{2}-25 y^{2}} \] Domain of \( z: 1 . I \quad 1 \)

To find the domain of the function \( z=\sqrt{400-16x^{2}-25y^{2}} \), we need the expression inside the square root to be non-negative. This means: \[ 400 - 16x^{2} - 25y^{2} \geq 0, \] which can be rearranged to: \[ 16x^{2} + 25y^{2} \leq 400. \] Dividing all terms by 400 gives us: \[ \frac{x^{2}}{25} + \frac{y^{2}}{16} \leq 1. \] This describes an ellipse centered at the origin with semi-major axis \( 5 \) along the \( x \)-axis (since \( \sqrt{25} = 5 \)) and semi-minor axis \( 4 \) along the \( y \)-axis (since \( \sqrt{16} = 4 \)). So the domain consists of all points \((x, y)\) that lie inside or on the boundary of this ellipse. The domain of \( z \) is thus: \[ \{(x, y) | \frac{x^{2}}{25} + \frac{y^{2}}{16} \leq 1\}. \] Just imagine backyard games where the players must stay within an elliptical boundary – that’s exactly what your domain looks like! No straying outside those limits! If you're curious about ellipses, consider diving into the properties of conic sections. These beautiful shapes are not just mathematical abstractions; they show up frequently in engineering, astronomy, and even art! Ellipses can describe planetary orbits and sound wave paths, truly intertwining math with the universe!