This is a simple inequality problem.
`h(t) = -16t^2+64t+4`
The requirement is `h(t)gt= 52` .
Muliplication by negative 1 changes the inequality.
Dividing by 16,
Now let's see how we can find the correct range.
There are three ranges.
`tlt=1` and `1lt=tlt=3` and `tgt=3`
Now let's substitute values in those ranges respectively and check for the sign. (The correct range should give negative as in the inequality)
`tlt=1` , Let's substitute t = 0,
`(t-3)(t-1) = (0-3)(0-1) = +3` positive
1<=t<=3, Let's substitute t = 2,
`(t-3)(t-1) = (2-3)(2-1) = -1`
This range give negative result.
t=>3, Let's substitute t = 4,
`(t-3)(t-1) = (4-3)(4-1) = +3` Positive.
Therefore the range which satisfies the inequality is,
Therefore between t =1 to t=3, the height of the ball is greater than or equal to 52.