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Answer :
To find out how long it takes for the fan to stop, we need to determine when the speed of the fan becomes zero. The speed of the fan in rotations per minute (rpm) is given by the equation:
[tex]\[ y = -5x^2 + 100x \][/tex]
where [tex]\( x \)[/tex] is the time in seconds and [tex]\( y \)[/tex] is the speed in rpm.
To find when the fan stops, we set [tex]\( y \)[/tex] to 0 because that represents the fan having stopped completely.
[tex]\[ 0 = -5x^2 + 100x \][/tex]
Now we need to solve this quadratic equation for [tex]\( x \)[/tex].
1. Factor the equation: Notice that both terms on the right-hand side have an [tex]\( x \)[/tex] in common, so we factor the equation:
[tex]\[ 0 = x(-5x + 100) \][/tex]
2. Set each factor to zero: This gives us two separate equations to solve:
- [tex]\( x = 0 \)[/tex]
- [tex]\(-5x + 100 = 0\)[/tex]
3. Solve each equation:
- The first equation, [tex]\( x = 0 \)[/tex], tells us that the fan starts at time [tex]\( x = 0 \)[/tex] seconds.
- For the second equation:
[tex]\[ -5x + 100 = 0 \][/tex]
Add 5x to both sides:
[tex]\[ 100 = 5x \][/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]. This means the fan starts at 0 seconds and stops completely at 20 seconds. Thus, it takes 20 seconds for the fan to stop spinning after being turned off.
[tex]\[ y = -5x^2 + 100x \][/tex]
where [tex]\( x \)[/tex] is the time in seconds and [tex]\( y \)[/tex] is the speed in rpm.
To find when the fan stops, we set [tex]\( y \)[/tex] to 0 because that represents the fan having stopped completely.
[tex]\[ 0 = -5x^2 + 100x \][/tex]
Now we need to solve this quadratic equation for [tex]\( x \)[/tex].
1. Factor the equation: Notice that both terms on the right-hand side have an [tex]\( x \)[/tex] in common, so we factor the equation:
[tex]\[ 0 = x(-5x + 100) \][/tex]
2. Set each factor to zero: This gives us two separate equations to solve:
- [tex]\( x = 0 \)[/tex]
- [tex]\(-5x + 100 = 0\)[/tex]
3. Solve each equation:
- The first equation, [tex]\( x = 0 \)[/tex], tells us that the fan starts at time [tex]\( x = 0 \)[/tex] seconds.
- For the second equation:
[tex]\[ -5x + 100 = 0 \][/tex]
Add 5x to both sides:
[tex]\[ 100 = 5x \][/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]. This means the fan starts at 0 seconds and stops completely at 20 seconds. Thus, it takes 20 seconds for the fan to stop spinning after being turned off.
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