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Answer :
To solve the problem of finding when the fan completely stops, we need to determine the times when the speed of the fan, modeled by the equation [tex]\( y = -5x^2 + 100x \)[/tex], is zero. This is because the fan stops completely when its speed is zero.
Here are the steps to find the solution:
1. Identify the Equation: We start with the equation representing the speed of the fan:
[tex]\[
y = -5x^2 + 100x
\][/tex]
2. Set the Speed to Zero: Since we want to find when the fan stops, we set [tex]\( y \)[/tex] equal to zero:
[tex]\[
0 = -5x^2 + 100x
\][/tex]
3. Factor the Equation: We factor out the common term, which is [tex]\( 5x \)[/tex], from the equation:
[tex]\[
0 = 5x(-x + 20)
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Set each factor equal to zero to solve for [tex]\( x \)[/tex]:
- For the first factor [tex]\( 5x = 0 \)[/tex], solving gives:
[tex]\[
x = 0
\][/tex]
- For the second factor [tex]\(-x + 20 = 0\)[/tex], solving gives:
[tex]\[
x = 20
\][/tex]
5. Interpret the Solutions: The solutions [tex]\( x = 0 \)[/tex] and [tex]\( x = 20 \)[/tex] represent the times, in seconds, at which the fan's speed is zero.
- At [tex]\( x = 0 \)[/tex], this is the moment when the fan is just starting (and hasn't sped up yet).
- At [tex]\( x = 20 \)[/tex], this is when the fan has returned to zero speed and has completely stopped.
Therefore, the fan completely stops after 20 seconds.
Here are the steps to find the solution:
1. Identify the Equation: We start with the equation representing the speed of the fan:
[tex]\[
y = -5x^2 + 100x
\][/tex]
2. Set the Speed to Zero: Since we want to find when the fan stops, we set [tex]\( y \)[/tex] equal to zero:
[tex]\[
0 = -5x^2 + 100x
\][/tex]
3. Factor the Equation: We factor out the common term, which is [tex]\( 5x \)[/tex], from the equation:
[tex]\[
0 = 5x(-x + 20)
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Set each factor equal to zero to solve for [tex]\( x \)[/tex]:
- For the first factor [tex]\( 5x = 0 \)[/tex], solving gives:
[tex]\[
x = 0
\][/tex]
- For the second factor [tex]\(-x + 20 = 0\)[/tex], solving gives:
[tex]\[
x = 20
\][/tex]
5. Interpret the Solutions: The solutions [tex]\( x = 0 \)[/tex] and [tex]\( x = 20 \)[/tex] represent the times, in seconds, at which the fan's speed is zero.
- At [tex]\( x = 0 \)[/tex], this is the moment when the fan is just starting (and hasn't sped up yet).
- At [tex]\( x = 20 \)[/tex], this is when the fan has returned to zero speed and has completely stopped.
Therefore, the fan completely stops after 20 seconds.
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