Vint is testing ceiling fans in a factory. For one of the tests, he switches the fan on, and after it attains a maximum speed of 500 rotations per minute (rpm), he switches the fan back off, recording the amount of time it takes for the fan to completely stop spinning. The given equation models Vint's test, where [tex]$x$[/tex] represents time in seconds and [tex]$y$[/tex] represents the speed in rotations per minute.

[tex]y = -5x^2 + 100x[/tex]

The equation has been graphed as shown.

To solve the problem, we need to find out when the ceiling fan comes to a complete stop. This means we need to determine the time [tex]$ x $[/tex] in seconds when the speed of the fan, [tex]$ y $[/tex], becomes zero.

The equation modeling the speed of the fan is:

[tex]\[ y = -5x^2 + 100x \][/tex]

Here, [tex]$ y $[/tex] represents the speed in rotations per minute (rpm), and [tex]$ x $[/tex] represents time in seconds.

To find out when the fan stops:

1. Set the equation equal to zero to represent the speed when the fan has stopped:

[tex]\[ -5x^2 + 100x = 0 \][/tex]

2. Factor the equation:

[tex]\[-5x(x - 20) = 0\][/tex]

3. Solve for [tex]$ x $[/tex] by setting each factor to zero:

[tex]\[
-5x = 0 \quad \Rightarrow \quad x = 0
\][/tex]

[tex]\[
x - 20 = 0 \quad \Rightarrow \quad x = 20
\][/tex]

4. Interpret the results:
- [tex]$ x = 0 $[/tex] is the time when the fan starts.
- [tex]$ x = 20 $[/tex] is the time when the fan stops completely.

Therefore, the fan completely stops spinning at 20 seconds.