Answer :
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.
[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.