To find the RPM at which the engine is putting out maximum horsepower, we need to determine the critical points of the power curve function and evaluate the horsepower at those points. The given power curve function is:
[tex]\[
y = -\frac{x^2}{30000} + \frac{11x}{25} - 11
\][/tex]
Step-by-step solution:
1. Find the derivative of the power curve function:
To find the critical points, we need to take the derivative of [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex]. The derivative [tex]\( \frac{dy}{dx} \)[/tex] will help us determine where the slope of the function is zero (indicating potential maxima or minima).
The original function is:
[tex]\[
y = -\frac{x^2}{30000} + \frac{11x}{25} - 11
\][/tex]
Taking the derivative with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{dy}{dx} = -\frac{2x}{30000} + \frac{11}{25}
\][/tex]
Simplify the derivative:
[tex]\[
\frac{dy}{dx} = -\frac{x}{15000} + \frac{11}{25}
\][/tex]
2. Set the derivative equal to zero and solve for [tex]\( x \)[/tex]:
Setting [tex]\( \frac{dy}{dx} \)[/tex] equal to zero to find the critical points:
[tex]\[
-\frac{x}{15000} + \frac{11}{25} = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
\frac{11}{25} = \frac{x}{15000}
\][/tex]
[tex]\[
x = 15000 \times \frac{11}{25}
\][/tex]
[tex]\[
x = 6600
\][/tex]
So, the RPM at which the engine is putting out maximum horsepower is:
[tex]\[
RPM = 6600
\][/tex]
3. Evaluate the horsepower at the critical point [tex]\( x = 6600 \)[/tex]:
Substitute [tex]\( x = 6600 \)[/tex] back into the original power curve function to find the horsepower:
[tex]\[
y = -\frac{6600^2}{30000} + \frac{11 \times 6600}{25} - 11
\][/tex]
Calculating step-by-step:
[tex]\[
y = -\frac{43560000}{30000} + \frac{72600}{25} - 11
\][/tex]
[tex]\[
y = -1452 + 2904 - 11
\][/tex]
[tex]\[
y = 1441
\][/tex]
So, the maximum horsepower is:
[tex]\[
HP = 1441
\][/tex]
Summary:
- The RPM at which the engine is putting out maximum horsepower is [tex]\( \boxed{6600} \)[/tex] RPM.
- The maximum horsepower is [tex]\( \boxed{1441} \)[/tex].
