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Select the correct answer.

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.

Answer :

To solve for the time it takes for the fan to completely stop spinning, we use the equation that models the fan's speed:

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

where [tex]\( y \)[/tex] is the speed in rotations per minute and [tex]\( x \)[/tex] is the time in seconds.

The fan stops spinning when the speed [tex]\( y \)[/tex] is 0. So, we need to find the value of [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex].

1. Set the equation equal to zero:
[tex]\[ 0 = -5x^2 + 100x \][/tex]

2. Factor the equation:
[tex]\[ 0 = x(-5x + 100) \][/tex]

3. Solve for [tex]\( x \)[/tex] by setting each factor equal to zero:

- First factor: [tex]\( x = 0 \)[/tex]
- Second factor: [tex]\( -5x + 100 = 0 \)[/tex]

4. Solve [tex]\( -5x + 100 = 0 \)[/tex]:
- Subtract 100 from both sides:
[tex]\[ -5x = -100 \][/tex]
- Divide both sides by -5:
[tex]\[ x = 20 \][/tex]

Therefore, the fan stops completely after 20 seconds. The solutions are [tex]\( x = 0 \)[/tex] and [tex]\( x = 20 \)[/tex]. Since [tex]\( x = 0 \)[/tex] represents the starting point when the fan is turned on, [tex]\( x = 20 \)[/tex] seconds is when the fan stops.