So we're given this position function. And first what we're doing with this position function is we're determining what the height is of the ball at its highest point. So in order to do this, we need to find where the velocity is equal to zero. Because velocity is equal to zero right about at this point because the derivative is zero the the slope of the tangent line zero. So this is really what we're looking after. We need to figure out at what time does this occur? And then we can put that time into the position function to determine the actual height. So, um, if we find the velocity of this function, we see that it is negative 32 t plus v. Not with that, um, the velocity. Now we just have to find where it's equal to zero. That's gonna be once again negative 32 t plus we not we saw for tea and see that t is going to be equal Teoh v not over 32. And we get that because we bring the 32 tea over here and then we divide by the 32 to get the T isolated So for Part B. Um, what we're going to see is that before we actually get to part B, what we have to dio is we have to now place this time into our original position function. So we now see if we evaluated at this point s a V not over a 32. This is going to equal negative 16 times. He not over 32 and that is squared, plus the not times e not over 32. And at this point, it's just simplifying to determine what this in fact is. And this is giving us the height at its maximum point. So we have 32 squared done there, and then we can cancel that out with the 16 to an extent. So it's gonna give us a negative the not squared over 64 and then So that way we can combine like terms. We're going to make this, um, 64 as well. So it's gonna us 64. And, Lord, to do that we have to double the numerator, so we will get to the not squared. And now that they have the same denominator, we can now say that the maximum height of this position function will be vey not squared, divided by 64. So that's part a, um now moving down to part B into it in green. So for part B were being asked more specifically to figure out the velocity with which the ball strikes the ground. So this we need to know when the ball strikes the ground the time at which the ball strikes the ground and when something strikes the ground, it means it has a height of zero. So what we need to figure out is, when is this going to equal zero? Okay, so we set zero equal to negative 16 t squared, plus veena t. What we can do here is we can factor out of tea. At this point. It just becomes pretty basic algebra. So I will give us a negative 16 t plus B not. And here we now have it equaling zero. So we see two cases we see t equals zero. Um, but with t equals zero, this doesn't really tell us anything when the time is equal to zero. This is the right before the ball is actually launched. So obviously the ball is not at any height because it has not yet been launched. So t equals zero we don't really care about because has nothing to do with the ball actually striking the ground because the ball hasn't even been launched yet. What we do care about is when we have zero equals negative 16 t plus me not because now we can solve for T. When we add 16 t to both sides and divide by 16 we see that t is equal to V not over 16. We're not quite done yet, though, because we just found the time in which it occurs. But what we really need to find is the velocity, um, of the ball when it strikes the ground. So we want to refer back to our velocity function, which is right here. Um, so we're gonna have v of the not over 16 and we're going to put this in place of tea when we're looking at our velocity function. So it was negative 32 t. So it is in fact, going to be negative 32 times being not over 16. And then that plus close Amina, right? And once again, just come just simple computing tells us that's going to be a negative to the not plus Vina, and that will simplify too negative v Not so. That is the velocity with which the ball actually strikes the ground.