Question

Can anyone please help me with this?

The elevation of a hiking trail is modelled by the function h(x)=2x^3+3x^2−17x+12 where h is the height in meters above sea level and x is the horizontal distance from a ranger station in kilometers. If x<0, the position is to the west of the station, and if x>0 the position is to the east. Since the trail extends 4.2km to the west of the ranger station and 4km to the east, the model is accurate for xεR. How can we determine which sections of the trail are above sea level?

Answer 1

a)
`x > 1.5`

b)
`-4 < x < 1`

Explanation:

Given that function is a third order polynomial, it can be factorized:

`h(x) = (x+4)\cdot (x-1.5)\cdot (x -1)`

The following inequation must be analyzed to determine which sections are above sea level:

`(x+4)\cdot (x-1.5)\cdot (x-1) >0`

By Algebra, it is known that product between three positive numbers or two negative numbers and a positive number are equal to a positive number. Then, there are only two trails:

a)
`x > 1.5`

b)
`-4 < x < 1`