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Answer :
To solve the problem, we need to understand the given quadratic equation that models the speed of the fan over time:
[tex]\[ y = -5x^2 + 100x \][/tex]
where [tex]\( y \)[/tex] is the speed in rotations per minute (rpm), and [tex]\( x \)[/tex] is the time in seconds.
The equation is a quadratic equation, which forms a parabola. Since the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-5\)[/tex]), the parabola opens downwards. This means the maximum point on the graph of this equation is at its vertex.
To find the time when the fan reaches its maximum speed, we need to find the x-coordinate of the vertex of the parabola. The formula to find the vertex of a quadratic equation given by [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In our equation, [tex]\( a = -5 \)[/tex] and [tex]\( b = 100 \)[/tex].
Substitute these values into the vertex formula:
[tex]\[ x = -\frac{100}{2 \times -5} \][/tex]
[tex]\[ x = -\frac{100}{-10} \][/tex]
[tex]\[ x = 10 \][/tex]
This means that after 10 seconds, the speed of the fan reaches its maximum.
Next, we substitute [tex]\( x = 10 \)[/tex] back into the original equation to find the maximum speed:
[tex]\[ y = -5(10)^2 + 100(10) \][/tex]
[tex]\[ y = -5(100) + 1000 \][/tex]
[tex]\[ y = -500 + 1000 \][/tex]
[tex]\[ y = 500 \][/tex]
Thus, the maximum speed of the fan is 500 rotations per minute, and it occurs 10 seconds after the fan is switched on. Therefore, the fan reaches its maximum speed of 500 rpm at 10 seconds.
[tex]\[ y = -5x^2 + 100x \][/tex]
where [tex]\( y \)[/tex] is the speed in rotations per minute (rpm), and [tex]\( x \)[/tex] is the time in seconds.
The equation is a quadratic equation, which forms a parabola. Since the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-5\)[/tex]), the parabola opens downwards. This means the maximum point on the graph of this equation is at its vertex.
To find the time when the fan reaches its maximum speed, we need to find the x-coordinate of the vertex of the parabola. The formula to find the vertex of a quadratic equation given by [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ x = -\frac{b}{2a} \][/tex]
In our equation, [tex]\( a = -5 \)[/tex] and [tex]\( b = 100 \)[/tex].
Substitute these values into the vertex formula:
[tex]\[ x = -\frac{100}{2 \times -5} \][/tex]
[tex]\[ x = -\frac{100}{-10} \][/tex]
[tex]\[ x = 10 \][/tex]
This means that after 10 seconds, the speed of the fan reaches its maximum.
Next, we substitute [tex]\( x = 10 \)[/tex] back into the original equation to find the maximum speed:
[tex]\[ y = -5(10)^2 + 100(10) \][/tex]
[tex]\[ y = -5(100) + 1000 \][/tex]
[tex]\[ y = -500 + 1000 \][/tex]
[tex]\[ y = 500 \][/tex]
Thus, the maximum speed of the fan is 500 rotations per minute, and it occurs 10 seconds after the fan is switched on. Therefore, the fan reaches its maximum speed of 500 rpm at 10 seconds.
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