Answer :
To solve the problem regarding the ceiling fan's speed over time, we need to understand the equation provided: [tex]\( y = -5x^2 + 100x \)[/tex]. In this equation, [tex]\( x \)[/tex] represents the time in seconds, and [tex]\( y \)[/tex] represents the speed of the fan in rotations per minute (rpm).
Let's break this down step-by-step:
1. Finding the Maximum Speed:
The equation [tex]\( y = -5x^2 + 100x \)[/tex] is a quadratic equation representing a parabola that opens downward (because the coefficient of [tex]\( x^2 \)[/tex] is negative). The highest point on this parabola will give us the maximum speed of the fan.
To find when this maximum speed occurs, we use the concept of taking the derivative. The first derivative of the equation with respect to [tex]\( x \)[/tex] will give us the rate of change of speed. Setting the derivative equal to zero will help us find the critical points.
From the calculations, the time at which the fan reaches maximum speed is at [tex]\( x = 10 \)[/tex] seconds.
Substituting [tex]\( x = 10 \)[/tex] back into the original equation gives us [tex]\( y = 500 \)[/tex]. Therefore, the maximum speed of the fan is 500 rpm.
2. Determining When the Fan Stops:
The fan comes to a complete stop when its speed [tex]\( y \)[/tex] equals 0. We set the equation [tex]\( y = -5x^2 + 100x \)[/tex] equal to 0 and solve for [tex]\( x \)[/tex].
Solving this equation shows that the fan stops at two points in time: [tex]\( x = 0 \)[/tex] seconds (the starting point when the fan is initially turned on) and [tex]\( x = 20 \)[/tex] seconds. Therefore, it takes 20 seconds for the fan to completely stop after reaching its maximum speed.
In summary:
- The fan reaches a maximum speed of 500 rpm after 10 seconds.
- It takes a total of 20 seconds for the fan to come to a complete stop from the starting point.
Let's break this down step-by-step:
1. Finding the Maximum Speed:
The equation [tex]\( y = -5x^2 + 100x \)[/tex] is a quadratic equation representing a parabola that opens downward (because the coefficient of [tex]\( x^2 \)[/tex] is negative). The highest point on this parabola will give us the maximum speed of the fan.
To find when this maximum speed occurs, we use the concept of taking the derivative. The first derivative of the equation with respect to [tex]\( x \)[/tex] will give us the rate of change of speed. Setting the derivative equal to zero will help us find the critical points.
From the calculations, the time at which the fan reaches maximum speed is at [tex]\( x = 10 \)[/tex] seconds.
Substituting [tex]\( x = 10 \)[/tex] back into the original equation gives us [tex]\( y = 500 \)[/tex]. Therefore, the maximum speed of the fan is 500 rpm.
2. Determining When the Fan Stops:
The fan comes to a complete stop when its speed [tex]\( y \)[/tex] equals 0. We set the equation [tex]\( y = -5x^2 + 100x \)[/tex] equal to 0 and solve for [tex]\( x \)[/tex].
Solving this equation shows that the fan stops at two points in time: [tex]\( x = 0 \)[/tex] seconds (the starting point when the fan is initially turned on) and [tex]\( x = 20 \)[/tex] seconds. Therefore, it takes 20 seconds for the fan to completely stop after reaching its maximum speed.
In summary:
- The fan reaches a maximum speed of 500 rpm after 10 seconds.
- It takes a total of 20 seconds for the fan to come to a complete stop from the starting point.
