Answer :
To find the engine speed that maximizes the torque, we need to consider the formula for torque given as:
[tex]\[ y = -3.75x^2 + 23.2x + 38.8 \][/tex]
where [tex]\( y \)[/tex] is the torque in foot-pounds, and [tex]\( x \)[/tex] is the speed of the engine in thousands of revolutions per minute (RPM).
The maximum torque occurs at a critical point of the function. Critical points can be found where the derivative of the function is zero or undefined.
Step-by-step solution:
1. Find the derivative of the torque function:
The given torque function is a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = -3.75 \)[/tex], [tex]\( b = 23.2 \)[/tex], and [tex]\( c = 38.8 \)[/tex]. The derivative, denoted as [tex]\( \frac{dy}{dx} \)[/tex], represents the rate of change of torque with respect to engine speed.
[tex]\[ \frac{dy}{dx} = \text{derivative of } -3.75x^2 + 23.2x + 38.8 \][/tex]
Using the power rule, the derivative is:
[tex]\[ \frac{dy}{dx} = -7.5x + 23.2 \][/tex]
2. Set the derivative to zero to find critical points:
To find where the torque is maximized or minimized, solve the equation:
[tex]\[ -7.5x + 23.2 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ -7.5x = -23.2 \][/tex]
Divide both sides by [tex]\(-7.5\)[/tex] to find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{23.2}{7.5} \][/tex]
Calculating this value gives:
[tex]\[ x \approx 3.093 \][/tex]
4. Interpret the result:
The engine speed that maximizes the torque is approximately 3.093 thousand RPMs, which is 3093 RPM.
Therefore, the engine speed that maximizes the torque is about 3093 RPM.
[tex]\[ y = -3.75x^2 + 23.2x + 38.8 \][/tex]
where [tex]\( y \)[/tex] is the torque in foot-pounds, and [tex]\( x \)[/tex] is the speed of the engine in thousands of revolutions per minute (RPM).
The maximum torque occurs at a critical point of the function. Critical points can be found where the derivative of the function is zero or undefined.
Step-by-step solution:
1. Find the derivative of the torque function:
The given torque function is a quadratic function of the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = -3.75 \)[/tex], [tex]\( b = 23.2 \)[/tex], and [tex]\( c = 38.8 \)[/tex]. The derivative, denoted as [tex]\( \frac{dy}{dx} \)[/tex], represents the rate of change of torque with respect to engine speed.
[tex]\[ \frac{dy}{dx} = \text{derivative of } -3.75x^2 + 23.2x + 38.8 \][/tex]
Using the power rule, the derivative is:
[tex]\[ \frac{dy}{dx} = -7.5x + 23.2 \][/tex]
2. Set the derivative to zero to find critical points:
To find where the torque is maximized or minimized, solve the equation:
[tex]\[ -7.5x + 23.2 = 0 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ -7.5x = -23.2 \][/tex]
Divide both sides by [tex]\(-7.5\)[/tex] to find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{23.2}{7.5} \][/tex]
Calculating this value gives:
[tex]\[ x \approx 3.093 \][/tex]
4. Interpret the result:
The engine speed that maximizes the torque is approximately 3.093 thousand RPMs, which is 3093 RPM.
Therefore, the engine speed that maximizes the torque is about 3093 RPM.
