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.

The equation [tex]y=-5x^2+100x[/tex] describes the relationship between time [tex]x[/tex] (in seconds) and the speed [tex]y[/tex] (in rpm) of the ceiling fan. This is a quadratic equation, and its graph is a downward-opening parabola.

Key Features of the Equation:

Maximum Speed:

The maximum speed occurs at the vertex of the parabola. The [tex]x[/tex]-coordinate of the vertex can be found using the formula:

[tex]x=-\frac{b}{2a}[/tex]

Here, [tex]a=-5[/tex] and [tex]b=100[/tex]:

[tex]x=-\frac{100}{2(-5)}=\frac{100}{10} =10[/tex]

Substituting [tex]x=10[/tex] back into the equation gives the maximum speed:

[tex]y=-5(10)^2+100(10)=-500+1000=500 rpm[/tex]

2. Initial Speed:

At [tex]x=0[/tex] (the moment the fan is turned on):

[tex]y=-5(0)^2+100(0)=0rpm[/tex]

3. Stopping Time:

To find when the fan stops completely, set [tex]y=0[/tex]:

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

Factor the equation:

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

This gives [tex]x=0[/tex] (when the fan is first turned on) and:

[tex]-5x+100=0[/tex] ---> [tex]5x=100[/tex] ---> [tex]x=20[/tex]

Therefore, the fan completely stops after 20 seconds.

Summary:

Maximum Speed: 500 rpm at [tex]x=10[/tex] seconds.
Fan Stops: The fan completely stops at [tex]x=20[/tex] seconds.