Given : When joey dives off a diving board, the equation of his pathway can be modeled by [tex]h=-16t^2+15t+12[/tex] .
We use vertex formula [tex]x=\dfrac{-b}{2a}[/tex] to find the maximum height of quadratic polynomial [tex]ax^2+bx+c[/tex]. (i)
As compare the given polynomial to (i), we have
a=-16 and b= 15
Then , the maximum height = [tex]\dfrac{-(15)}{2(-16)}=\dfrac{15}{32}=0.46875[/tex]
Hence, the maximum height is 0.46875 m.
When Joey reaches the water then h= 0
i.e. [tex]0=-16t^2+15t+12[/tex]
[tex]16t^2-15t-12=0\\\\ t=\dfrac{-(-15)\pm\sqrt{(-15)^2-4(16)(-12)}}{2(16)}\\\\=\dfrac{15\pm\sqrt{225+768}}{32}=\dfrac{15\pm\sqrt{993} }{32}=\dfrac{15\pm31.512}{32}\\\\t=\dfrac{15-31.512}{32}\ \ or\ \ t=\dfrac{15+31.512}{32}\\\\t=-0.516\ or \ t=1.4535\approx1.45\\\\\Rightarrow\ t=1.45\ \ \ [\text{Time cannot be negative}][/tex]
Hence, the time it will take for Joey to reach the water is 1.45 seconds.
