how do you figure out how many seconds an object takes to reach its maximum height from the equation s(t)=16t(squared)+176t+64?

My first reaction was to suggest you take the first derivative of s(t) and set it equal to zero -- this will tell you at what time t the velocity of s is zero. That will be the moment when the object reaches its maximum altitude. However, if you try that you will see that s'(t) can only be zero for a negative value of t, which means that from t=0 onward the object continues to accelerate. Are you sure you have the equation for s(t) correct? Looks to me like perhaps there should be a negative sign in front of the 16t^2 term?