The shape of a dome can be modeled by the equation $h=-2 d^{2}+100$ where $h$ is the height (in feet) of th dome from the floor $d$ feet from its center. How far from the center of the dome is the height 50 feet?

Step 1 ：The shape of a dome can be modeled by the equation \(h=-2 d^{2}+100\) where \(h\) is the height (in feet) of the dome from the floor \(d\) feet from its center. We are asked to find how far from the center of the dome is the height 50 feet.

Step 2 ：This means we need to solve the equation \(h=-2 d^{2}+100\) for \(d\) when \(h=50\). This is a quadratic equation, so we can use the quadratic formula to solve for \(d\). The quadratic formula is \(d=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation.

Step 3 ：In this case, \(a=-2\), \(b=0\), and \(c=100-h\). We need to substitute \(h=50\) into the equation and solve for \(d\).

Step 4 ：The solutions to the equation are \(d=-5.0\) and \(d=5.0\). However, distance cannot be negative, so we discard the negative solution.

Step 5 ：Therefore, the distance from the center of the dome when the height is 50 feet is 5.0 feet.

Step 6 ：Final Answer: The distance from the center of the dome when the height is 50 feet is \(\boxed{5.0}\) feet.