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Answer :
To find the engine speed that maximizes the torque, we need to determine the vertex of the parabola represented by the quadratic equation [tex]\( y = -3.75x^2 + 23.2x + 38.8 \)[/tex].
This equation is in the standard form of a quadratic: [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -3.75 \)[/tex]
- [tex]\( b = 23.2 \)[/tex]
- [tex]\( c = 38.8 \)[/tex]
The vertex of a parabola given by a quadratic equation can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
This formula helps us find the speed (x) at which the torque (y) is maximized, because the parabola opens downwards (since [tex]\( a \)[/tex] is negative).
Let's plug in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into this formula:
[tex]\[ x = -\frac{23.2}{2 \times (-3.75)} \][/tex]
Calculating the value:
[tex]\[ x = -\frac{23.2}{-7.5} \][/tex]
[tex]\[ x \approx 3.0933 \][/tex]
Therefore, the engine speed that maximizes the torque is approximately 3.09 (rounded to two decimal places). This is the optimal speed at which the engine torque is at its maximum.
This equation is in the standard form of a quadratic: [tex]\( y = ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -3.75 \)[/tex]
- [tex]\( b = 23.2 \)[/tex]
- [tex]\( c = 38.8 \)[/tex]
The vertex of a parabola given by a quadratic equation can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
This formula helps us find the speed (x) at which the torque (y) is maximized, because the parabola opens downwards (since [tex]\( a \)[/tex] is negative).
Let's plug in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into this formula:
[tex]\[ x = -\frac{23.2}{2 \times (-3.75)} \][/tex]
Calculating the value:
[tex]\[ x = -\frac{23.2}{-7.5} \][/tex]
[tex]\[ x \approx 3.0933 \][/tex]
Therefore, the engine speed that maximizes the torque is approximately 3.09 (rounded to two decimal places). This is the optimal speed at which the engine torque is at its maximum.
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