Answer :
To solve this problem, we need to analyze the given equation that models the speed of a ceiling fan during Vint's test. The equation provided is:
[tex]\[ y = -5x^2 + 100x \][/tex]
where [tex]\( y \)[/tex] represents the speed of the fan in rotations per minute (rpm), and [tex]\( x \)[/tex] represents time in seconds.
Step-by-step Solution:
1. Understanding the Equation:
- The equation is a quadratic function and represents a parabola that opens downwards (because the coefficient of [tex]\( x^2 \)[/tex] is negative).
2. Finding Key Features:
- Vertex: The vertex of this parabola will give us the time at which the fan reaches its maximum speed, which is the peak of the graph.
- Intercepts: Finding the x-intercepts will tell us when the fan speed is zero, i.e., when it starts and stops.
3. Finding the Vertex:
- The vertex of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
- Here, [tex]\( a = -5 \)[/tex], [tex]\( b = 100 \)[/tex]. Substituting these values in:
[tex]\[
x = -\frac{100}{2 \times (-5)} = \frac{100}{10} = 10
\][/tex]
- At [tex]\( x = 10 \)[/tex] seconds, the fan reaches its maximum speed. To find that maximum speed, substitute [tex]\( x = 10 \)[/tex] back into the original equation:
[tex]\[
y = -5(10)^2 + 100(10) = -500 + 1000 = 500 \text{ rpm}
\][/tex]
- This confirms the maximum speed of the fan is 500 rpm, which is consistent with the problem statement.
4. Finding the x-intercepts:
- We solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex] to find out when the fan stops spinning.
[tex]\[
-5x^2 + 100x = 0
\][/tex]
- Factor the equation:
[tex]\[
x(-5x + 100) = 0
\][/tex]
- This gives us [tex]\( x = 0 \)[/tex] or [tex]\( -5x + 100 = 0 \)[/tex].
- Solving [tex]\( -5x + 100 = 0 \)[/tex] gives:
[tex]\[
-5x = -100 \Rightarrow x = 20
\][/tex]
- So, the fan stops spinning at [tex]\( x = 20 \)[/tex] seconds.
5. Conclusion:
- The fan reaches its maximum speed of 500 rpm at 10 seconds after being turned on.
- It comes to a complete stop at 20 seconds.
This detailed analysis shows the behavior of the fan during the test, aligning with the model's description using the given quadratic equation.
[tex]\[ y = -5x^2 + 100x \][/tex]
where [tex]\( y \)[/tex] represents the speed of the fan in rotations per minute (rpm), and [tex]\( x \)[/tex] represents time in seconds.
Step-by-step Solution:
1. Understanding the Equation:
- The equation is a quadratic function and represents a parabola that opens downwards (because the coefficient of [tex]\( x^2 \)[/tex] is negative).
2. Finding Key Features:
- Vertex: The vertex of this parabola will give us the time at which the fan reaches its maximum speed, which is the peak of the graph.
- Intercepts: Finding the x-intercepts will tell us when the fan speed is zero, i.e., when it starts and stops.
3. Finding the Vertex:
- The vertex of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
- Here, [tex]\( a = -5 \)[/tex], [tex]\( b = 100 \)[/tex]. Substituting these values in:
[tex]\[
x = -\frac{100}{2 \times (-5)} = \frac{100}{10} = 10
\][/tex]
- At [tex]\( x = 10 \)[/tex] seconds, the fan reaches its maximum speed. To find that maximum speed, substitute [tex]\( x = 10 \)[/tex] back into the original equation:
[tex]\[
y = -5(10)^2 + 100(10) = -500 + 1000 = 500 \text{ rpm}
\][/tex]
- This confirms the maximum speed of the fan is 500 rpm, which is consistent with the problem statement.
4. Finding the x-intercepts:
- We solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex] to find out when the fan stops spinning.
[tex]\[
-5x^2 + 100x = 0
\][/tex]
- Factor the equation:
[tex]\[
x(-5x + 100) = 0
\][/tex]
- This gives us [tex]\( x = 0 \)[/tex] or [tex]\( -5x + 100 = 0 \)[/tex].
- Solving [tex]\( -5x + 100 = 0 \)[/tex] gives:
[tex]\[
-5x = -100 \Rightarrow x = 20
\][/tex]
- So, the fan stops spinning at [tex]\( x = 20 \)[/tex] seconds.
5. Conclusion:
- The fan reaches its maximum speed of 500 rpm at 10 seconds after being turned on.
- It comes to a complete stop at 20 seconds.
This detailed analysis shows the behavior of the fan during the test, aligning with the model's description using the given quadratic equation.
