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.
[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.
