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} (dy)/(dx)=-(2x)/(15000)+(27)/(44) \\ (d^(2)y)/(dx^(2))=-(2)/(15000) \end{gathered}`

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

`\begin{gathered} -(2x)/(15000)+(27)/(44)=0 \\ (2x)/(15000)=(27)/(44) \\ x=(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=-((4602.273)^(2))/(15000)+(27)/(44)(4602.273)-14 \\ y=1398.061 \end{gathered}`

Therefore, the maximum horsepower is 1398.061.