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Answer :
To solve the problem, we're working with the equation that models the speed of the fan, given by [tex]\( y = -5x^2 + 100x \)[/tex], where [tex]\( y \)[/tex] represents the speed in rotations per minute (rpm) and [tex]\( x \)[/tex] is the time in seconds.
The goal is to find out when the fan reaches its maximum speed of 500 rpm and when it completely stops.
1. Find the critical points:
To find when the fan reaches its maximum speed, we first need to find the critical points of the equation. This is done by taking the derivative of the speed equation with respect to time [tex]\( x \)[/tex] and setting it to zero to find where the rate of change switches direction.
- The derivative [tex]\( y' \)[/tex] of the equation [tex]\( y = -5x^2 + 100x \)[/tex] is:
[tex]\[
y' = \frac{d}{dx}(-5x^2 + 100x) = -10x + 100
\][/tex]
- Set the derivative equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[
-10x + 100 = 0
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
-10x + 100 = 0 \\
10x = 100 \\
x = 10
\][/tex]
Therefore, the critical point is at [tex]\( x = 10 \)[/tex] seconds.
2. Determine the speed at the critical point:
Substitute [tex]\( x = 10 \)[/tex] back into the original equation to find the speed at that time.
[tex]\[
y = -5(10)^2 + 100(10) \\
y = -5(100) + 1000 \\
y = -500 + 1000 \\
y = 500
\][/tex]
When [tex]\( x = 10 \)[/tex] seconds, the speed of the fan is indeed 500 rpm, indicating that the fan reaches its maximum speed at this point.
3. Find when the fan stops:
To find when the fan completely stops, we need to solve for [tex]\( x \)[/tex] when the speed [tex]\( y = 0 \)[/tex].
- Set the original equation to zero:
[tex]\[
-5x^2 + 100x = 0
\][/tex]
- Factor the equation:
[tex]\[
x(-5x + 100) = 0
\][/tex]
- Solving the factored equation, we get:
[tex]\[
x = 0 \quad \text{or} \quad -5x + 100 = 0
\][/tex]
- Solving for the second condition:
[tex]\[
-5x + 100 = 0 \\
5x = 100 \\
x = 20
\][/tex]
So, the fan completely stops at [tex]\( x = 20 \)[/tex] seconds.
In conclusion, the fan reaches its maximum speed of 500 rpm at [tex]\( x = 10 \)[/tex] seconds and it stops at [tex]\( x = 20 \)[/tex] seconds.
The goal is to find out when the fan reaches its maximum speed of 500 rpm and when it completely stops.
1. Find the critical points:
To find when the fan reaches its maximum speed, we first need to find the critical points of the equation. This is done by taking the derivative of the speed equation with respect to time [tex]\( x \)[/tex] and setting it to zero to find where the rate of change switches direction.
- The derivative [tex]\( y' \)[/tex] of the equation [tex]\( y = -5x^2 + 100x \)[/tex] is:
[tex]\[
y' = \frac{d}{dx}(-5x^2 + 100x) = -10x + 100
\][/tex]
- Set the derivative equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[
-10x + 100 = 0
\][/tex]
- Solving for [tex]\( x \)[/tex]:
[tex]\[
-10x + 100 = 0 \\
10x = 100 \\
x = 10
\][/tex]
Therefore, the critical point is at [tex]\( x = 10 \)[/tex] seconds.
2. Determine the speed at the critical point:
Substitute [tex]\( x = 10 \)[/tex] back into the original equation to find the speed at that time.
[tex]\[
y = -5(10)^2 + 100(10) \\
y = -5(100) + 1000 \\
y = -500 + 1000 \\
y = 500
\][/tex]
When [tex]\( x = 10 \)[/tex] seconds, the speed of the fan is indeed 500 rpm, indicating that the fan reaches its maximum speed at this point.
3. Find when the fan stops:
To find when the fan completely stops, we need to solve for [tex]\( x \)[/tex] when the speed [tex]\( y = 0 \)[/tex].
- Set the original equation to zero:
[tex]\[
-5x^2 + 100x = 0
\][/tex]
- Factor the equation:
[tex]\[
x(-5x + 100) = 0
\][/tex]
- Solving the factored equation, we get:
[tex]\[
x = 0 \quad \text{or} \quad -5x + 100 = 0
\][/tex]
- Solving for the second condition:
[tex]\[
-5x + 100 = 0 \\
5x = 100 \\
x = 20
\][/tex]
So, the fan completely stops at [tex]\( x = 20 \)[/tex] seconds.
In conclusion, the fan reaches its maximum speed of 500 rpm at [tex]\( x = 10 \)[/tex] seconds and it stops at [tex]\( x = 20 \)[/tex] seconds.
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