Step-by-step explanation:
Since the initial velocity of the object is 170 ft/s, the expression for h is given by
[tex]h = -16t^2 + 170t[/tex]
In order to find the time it takes for the object to reach the height of 302 ft, we to rewrite the equation above as
[tex]302 = -16t^2 +170t \Rightarrow 16t^2 - 170t + 302 = 0[/tex]
This is a quadratic equation whose roots are
[tex]t = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
where a = 16, b = -170 and c = 302. Using these values, we get
[tex]t = \dfrac{170 \pm \sqrt{(-170)^2 - 4(16)(302)}}{2(16)}[/tex]
[tex]\;\;\;=\dfrac{170 \pm \sqrt{28900 - 19328}}{32}[/tex]
[tex]\;\;\;= \dfrac{170 \pm 97.8}{32}[/tex]
[tex]\;\;\;= 2.3\:\text{s},\;\;8.4\:\text{s},[/tex]
This means the object will reach the height of 302 ft 2.3 seconds after launch and then at 8.4 seconds after launch (on its way down).
