The height of a small rise in a roller coaster track is modeled by () = −. + . + . , where x is the distance in feet from a supported pole at ground level. Find the greatest height of the rise.

Maximum height is 7 feet

Step-by-step explanation:

Solution:-

- The complete question is as follows:

" The height of a small rise in a roller coaster track is modeled by f(x) = –0.07x^2 + 0.42x + 6.37, where x is the distance in feet from a supported at ground level.

Find the greatest height of the rise "

- To find any turning points ( minimum or maximum ) points of a trajectory expressed as function of independent parameter, we find the critical points of the trajectory where the first derivative of the dependent variable w.rt independent variable is set to zero.

- In our case the height of the roller coaster track (y) is function of the distance (x) from a supported pole at ground level.

                    f(x) = –0.07x^2 + 0.42x + 6.37

- Now set the first derivative equal to zero, and determine the critical values of x:

                    0 =  -0.14x + 0.42

                    x = 0.42 / 0.14 = 3 ft

- The critical value for the coaster track is at point 3 feet away from the supported pole at ground level. So the height f(x) at x = 3 ft, would be:

                    f ( x = 3 ) = max height

                    max height =  –0.07*3^2 + 0.42*3 + 6.37

                                        = 7 ft