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.
[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.
