I assume you meant h(t) = 160t - 16t^2, as h(t) = 160t - 16t2 has no maximum.
There's two ways of solving this:
(i) By completing the square and finding the maximum turning point.
(ii) By using Calculus methods to find derivative and equating it to 0 in order to find maximum turning point.
(i) h(t) = 160t - 16t^2
h(t) = -16t^2 +160t
h(t) = -16(t^2-10t) (by taking out a common factor of -16)
h(t) = -16[(t-5)^2 - 25] (by completing the square)
h(t) = -16(t-5)^2 + 400 (on multiplying out by -16)
From this we see that the turning point is at (5;400), therefore the maximum height is 400 feet and is reached after 5 seconds.
Or...
(ii) h(t) = 160t - 16t^2
h`(t) = 160 - 32t (where h`(t) = derivative of h(t))
Now to find maximum of h(t), we set h`(t) = 0 and solve for t:
0 = 160 - 32t (on substituting h`(t) = 0)
32t = 160 (on solving for t)
t = 160/32 (on dividing both sides by 5)
t= 5
Now we have found that at 5 seconds, we will reach our maximum height. So to find this maximum height, we'll have to substitute t=5 into h(t) = 160t - 16t^2.
h(5) = 160(5) -16(5)^2
h(5) = 800 - 400
h(5) = 400 feet
So, once again we have shown that maximum height is 400 feet and is reached after 5 seconds.
