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Answer :
To solve the problem of determining how long it takes for the fan to stop completely, we need to analyze the given equation, which describes the speed (in rotations per minute) of the fan as a function of time (in seconds):
[tex]\[
y = -5x^2 + 100x
\][/tex]
Here, [tex]\( y \)[/tex] represents the speed in rotations per minute, and [tex]\( x \)[/tex] represents the time in seconds. We need to find out when the speed (y) becomes zero since that indicates the fan has completely stopped.
Step-by-step solution:
1. Set the speed equation to zero: Since we are interested in the time when the fan stops completely, we need to find when [tex]\( y = 0 \)[/tex]. So, set the equation equal to zero:
[tex]\[
-5x^2 + 100x = 0
\][/tex]
2. Factor the equation: To solve this quadratic equation, factor out the common term:
[tex]\[
x(-5x + 100) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]: Use the zero-product property, which states that if a product of two factors is zero, at least one of the factors must be zero. This gives us two equations to solve:
[tex]\[
x = 0 \quad \text{or} \quad -5x + 100 = 0
\][/tex]
4. Solve the first equation:
[tex]\[
x = 0
\][/tex]
This solution indicates the initial start time when the fan begins operating.
5. Solve the second equation:
[tex]\[
-5x + 100 = 0
\][/tex]
[tex]\[
-5x = -100
\][/tex]
[tex]\[
x = \frac{-100}{-5}
\][/tex]
[tex]\[
x = 20
\][/tex]
The solutions to [tex]\( x \)[/tex] are [tex]\( x = 0 \)[/tex] and [tex]\( x = 20 \)[/tex]. The fan starts at [tex]\( x = 0 \)[/tex] and stops completely at [tex]\( x = 20 \)[/tex] seconds.
Conclusion: The entire test from when the fan is turned on to when it completely stops takes 20 seconds.
[tex]\[
y = -5x^2 + 100x
\][/tex]
Here, [tex]\( y \)[/tex] represents the speed in rotations per minute, and [tex]\( x \)[/tex] represents the time in seconds. We need to find out when the speed (y) becomes zero since that indicates the fan has completely stopped.
Step-by-step solution:
1. Set the speed equation to zero: Since we are interested in the time when the fan stops completely, we need to find when [tex]\( y = 0 \)[/tex]. So, set the equation equal to zero:
[tex]\[
-5x^2 + 100x = 0
\][/tex]
2. Factor the equation: To solve this quadratic equation, factor out the common term:
[tex]\[
x(-5x + 100) = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]: Use the zero-product property, which states that if a product of two factors is zero, at least one of the factors must be zero. This gives us two equations to solve:
[tex]\[
x = 0 \quad \text{or} \quad -5x + 100 = 0
\][/tex]
4. Solve the first equation:
[tex]\[
x = 0
\][/tex]
This solution indicates the initial start time when the fan begins operating.
5. Solve the second equation:
[tex]\[
-5x + 100 = 0
\][/tex]
[tex]\[
-5x = -100
\][/tex]
[tex]\[
x = \frac{-100}{-5}
\][/tex]
[tex]\[
x = 20
\][/tex]
The solutions to [tex]\( x \)[/tex] are [tex]\( x = 0 \)[/tex] and [tex]\( x = 20 \)[/tex]. The fan starts at [tex]\( x = 0 \)[/tex] and stops completely at [tex]\( x = 20 \)[/tex] seconds.
Conclusion: The entire test from when the fan is turned on to when it completely stops takes 20 seconds.
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