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Answer :
We begin with the model for the fan test:
[tex]$$
y = -5x^2 + 100x,
$$[/tex]
where [tex]$x$[/tex] is the time in seconds and [tex]$y$[/tex] is the speed in rotations per minute.
Step 1. Set the speed equal to zero.
When the fan stops, [tex]$y = 0$[/tex], so:
[tex]$$
-5x^2 + 100x = 0.
$$[/tex]
Step 2. Factor the equation.
We factor out the common term [tex]$-5x$[/tex]:
[tex]$$
-5x(x - 20) = 0.
$$[/tex]
The equation is now already separated into factors.
Step 3. Solve for [tex]$x$[/tex].
For the product to be zero, either factor must be zero:
1. [tex]$-5x = 0$[/tex] leads to [tex]$x = 0$[/tex],
2. [tex]$x - 20 = 0$[/tex] leads to [tex]$x = 20$[/tex].
Since [tex]$x = 0$[/tex] corresponds to the initial moment when the fan is turned on, we ignore it. The meaningful solution is:
[tex]$$
x = 20.
$$[/tex]
Step 4. Interpret the result.
The fan comes to a complete stop after 20 seconds.
For completeness, the quadratic equation has a discriminant given by:
[tex]$$
D = b^2 - 4ac,
$$[/tex]
where [tex]$a = -5$[/tex], [tex]$b = 100$[/tex], and [tex]$c = 0$[/tex]. After calculation:
[tex]$$
D = 100^2 - 4(-5)(0) = 10000.
$$[/tex]
The roots obtained using the quadratic formula confirm the times [tex]$x=0$[/tex] and [tex]$x=20$[/tex] seconds.
Conclusion:
The fan takes [tex]$\boxed{20}$[/tex] seconds to completely stop.
[tex]$$
y = -5x^2 + 100x,
$$[/tex]
where [tex]$x$[/tex] is the time in seconds and [tex]$y$[/tex] is the speed in rotations per minute.
Step 1. Set the speed equal to zero.
When the fan stops, [tex]$y = 0$[/tex], so:
[tex]$$
-5x^2 + 100x = 0.
$$[/tex]
Step 2. Factor the equation.
We factor out the common term [tex]$-5x$[/tex]:
[tex]$$
-5x(x - 20) = 0.
$$[/tex]
The equation is now already separated into factors.
Step 3. Solve for [tex]$x$[/tex].
For the product to be zero, either factor must be zero:
1. [tex]$-5x = 0$[/tex] leads to [tex]$x = 0$[/tex],
2. [tex]$x - 20 = 0$[/tex] leads to [tex]$x = 20$[/tex].
Since [tex]$x = 0$[/tex] corresponds to the initial moment when the fan is turned on, we ignore it. The meaningful solution is:
[tex]$$
x = 20.
$$[/tex]
Step 4. Interpret the result.
The fan comes to a complete stop after 20 seconds.
For completeness, the quadratic equation has a discriminant given by:
[tex]$$
D = b^2 - 4ac,
$$[/tex]
where [tex]$a = -5$[/tex], [tex]$b = 100$[/tex], and [tex]$c = 0$[/tex]. After calculation:
[tex]$$
D = 100^2 - 4(-5)(0) = 10000.
$$[/tex]
The roots obtained using the quadratic formula confirm the times [tex]$x=0$[/tex] and [tex]$x=20$[/tex] seconds.
Conclusion:
The fan takes [tex]$\boxed{20}$[/tex] seconds to completely stop.
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