To find the height you only have to substitute tt in the equation for the number of seconds you what to know its possition,
d(t)=-16·1^2+30·1+6=20d(t)=−16⋅12+30⋅1+6=20 mm
and to find the time it took the hat to hit the ground you need to think a bit. If the hat is in the ground it means that de height is 00, so you can equal the equation to 00 and find the value of tt.
-16t^2+30t+6=0−16t2+30t+6=0
we can use the formula for quadratics equations.
t=(-b+-sqrt(b^2-4ac))/(2a)=(-30+-sqrt(30^2-4·(-16)·6))/(2·(-16))=t=−b±√b2−4ac2a=−30±√302−4⋅(−16)⋅62⋅(−16)=
=(-30+-sqrt(900+384))/(-32)=(-30+-sqrt(1284))/(-32)=−30±√900+384−32=−30±√1284−32
cancel(t_1=(-30+sqrt(1284))/(-32)=-0.18s)
This solution is not possible because time can't be negative.
t_2=(-30-sqrt(1284))/(-32)=2.06 s
