Question 140798
PART ONE
{{{s= -16t^2+v[0]t+s[0]}}}
{{{s(0.5)= -16(0.5)^2+32(0.5)+40}}}
{{{s(0.5)= -16(0.25)+16+40}}}
{{{s(0.5)= -4+16+40}}}
{{{s(0.5)= 52}}}
Yes, you are correct.
52 feet is the correct height.
PART TWO

I'm not sure how advanced you are in mathematics to solve the next part. 
If you have taken calculus, you take the derivative of the function and set it equal to zero. 
If you are using algebra, you can recognize that the equation is the equation of a parabola, find the axis of symmetry, and calculate the maximum there.
Finally you could plot points and try to find the maximum that way. 
We can do all three.
1.{{{s= -16t^2+32t+40}}}
{{{ds/dt=-32t+32=0}}}
{{{-32t+32=0}}}
{{{t=1}}}
The maximum occurs at t=1 second.
{{{s(1)=-16(1)+32(1)+40}}}
{{{s(1)=56}}}
2.{{{s= -16t^2+32t+40}}}
The axis of symmetry for the parabola 
{{{ax^2+bx+c=0}}} occurs at {{{x=-b/2a}}}
In your case,
a=-16
b=32
The axis of symmetry is then 
{{{t=-32/2(-16)}}}
{{{t=1}}}
Just as above, s=56 ft at t=1 sec.
3. Start at t=0 and solve for s.
(0.0,40)
(0.5,52)
(1.0,56)
(1.5,52)
(2.0,40)
You could also take some points aroung t=1 to prove to yourself that the maximum does occur at t=1.