Question

The engine in a airplane has a power curve approximated by x2 270 y = + 14 15000 44 where I is the RPin and y is the horsepower generated. - At what RPM is the engine putting out maximum horsepower?. Round your answer to three decimal places. RPM What is the maximum horsepower? Round your answer to three decimal places.

Answer 1

A)For this question, we will use the first and second derivative criteria. First, we compute the first and second derivative of the given function:

$\begin{gathered} \frac{dy}{dx}=-\frac{2x}{15000}+\frac{27}{44} \\ \frac{d^{2}y}{dx^{2}}=-\frac{2}{15000} \end{gathered}$

Setting the first derivative equal to zero and solving for x we get:

$\begin{gathered} -\frac{2x}{15000}+\frac{27}{44}=0 \\ \frac{2x}{15000}=\frac{27}{44} \\ x=\frac{27(15000)}{44(2)}=4602.273 \end{gathered}$

Evaluating the first second derivative at x=4602.273 we get a negative number, therefore the function has a maximum value at x=4602.273. At 4602.273 RPM the engine puts out its maximum horsepower.

B) Now, to compute the maximum horsepower we evaluate the given function at x=4602.273:

$\begin{gathered} y=-\frac{(4602.273)^{2}}{15000}+\frac{27}{44}(4602.273)-14 \\ y=1398.061 \end{gathered}$

Therefore, the maximum horsepower is 1398.061.