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Vint is testing ceiling fans in a factory. For one of the tests, he switches the fan on, and after it attains a maximum speed of 500 rotations per minute (rpm), he switches the fan back off, recording the amount of time it takes for the fan to completely stop spinning. The given equation models Vint's test, where [tex]x[/tex] represents time in seconds and [tex]y[/tex] represents the speed in rotations per minute:

[tex]\[ y = -5x^2 + 100x \][/tex]

The equation has been graphed as shown.

Answer :

To solve the problem of finding when the fan reaches its maximum speed of 500 rotations per minute (rpm), we need to analyze the given equation:

[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.

The task is to determine when the speed [tex]\( y \)[/tex] reaches 500 rpm. Thus, we set up the equation:

[tex]\[ -5x^2 + 100x = 500 \][/tex]

Let's solve this step-by-step:

1. Rearrange the Equation:
We bring all terms to one side to set the equation to zero:

[tex]\[ -5x^2 + 100x - 500 = 0 \][/tex]

2. Identify the Coefficients:
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = -5 \)[/tex]
- [tex]\( b = 100 \)[/tex]
- [tex]\( c = -500 \)[/tex]

3. Use the Quadratic Formula:
The quadratic formula to find the roots of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

4. Calculate the Discriminant:
The discriminant [tex]\((b^2 - 4ac)\)[/tex] determines the number and type of roots:

[tex]\[ \text{Discriminant} = 100^2 - 4 \times (-5) \times (-500) \][/tex]

[tex]\[ \text{Discriminant} = 10000 - 10000 = 0 \][/tex]

5. Find the Roots:
Since the discriminant is zero, there is one real root (meaning [tex]\( x \)[/tex] has one unique solution):

[tex]\[ x = \frac{-100 \pm \sqrt{0}}{2 \times -5} \][/tex]

[tex]\[ x = \frac{-100}{-10} \][/tex]

[tex]\[ x = 10 \][/tex]

6. Conclusion:
The fan reaches its maximum speed of 500 rpm at 10 seconds. Therefore, the time it takes for the fan to reach this speed is 10 seconds.