At what RPM is engine putting out max horsepower algebra problem

Question

The engine in a ice cream truck has a power curve approximated by Y=-X^2/5000 + 19x/23 -26

X is the RPM, Y is the horsepower generated (notice the a and b values are fractions)

At what RPM is the engine putting out maximum horsepower?. Round your answer to three decimal places.

RPM___

Answer 1

y = -ax^2 + bx + c is a parabola that opens down. The maximum is the vertex which is -b/2a

For Y= -(X^2)/5000 + 19x/23 - 26 a = -1/5000 and b = 19/23

-b/2a = -19/23 / 2(-1/5000) = -19/23 / -2/5000 = -19/23 / -1/2500

= -19/23 * -2500/1 = about 2065.217. That's the x value of the maximum, and x is the RPM at max.

Max RPM is about 2065.217