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 determine the time it takes for the fan to stop spinning, we need to find out when the speed [tex]$ y $[/tex] becomes zero in the equation:

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

where [tex]$ x $[/tex] is the time in seconds.

To find the time when the speed is zero, we set [tex]$ y = 0 $[/tex] and solve for [tex]$ x $[/tex]:

1. Set the equation to zero:

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

2. Factor the quadratic equation:

[tex]\[ 0 = x(-5x + 100) \][/tex]

3. Set each factor equal to zero to find the values of [tex]$ x $[/tex]:

- For the first factor:
[tex]\[ x = 0 \][/tex]

- For the second factor:
[tex]\[ -5x + 100 = 0 \][/tex]

4. Solve the second equation:

[tex]\[ -5x = -100 \][/tex]

Divide both sides by -5:

[tex]\[ x = 20 \][/tex]

The solutions to the equation are [tex]$ x = 0 $[/tex] and [tex]$ x = 20 $[/tex]. These values of [tex]$ x $[/tex] are the times in seconds at which the speed of the fan is zero.

This means the fan starts at [tex]$ x = 0 $[/tex] and comes to a complete stop at [tex]$ x = 20 $[/tex] seconds. Therefore, it takes 20 seconds for the fan to stop spinning.