 In this video we provide the solution to question number 12 for practice exam number 4 for Maths 12-10. We're given the velocity function of a falling ball. It's given as v of t equals negative 9.8 t minus 25 meters per second. And we're also given the initial height of the ball as 1000 meters. So how high is the ball after 10 seconds? So we need to figure out what is the position function, knowing that velocity is the derivative position. We need to take the anti-derivative here. So our position function s of t, this is going to equal the anti-derivative of v of t with respect to time here. So we need to find the anti-derivative of a negative 9.8 t minus 25 dt. And we can calculate this using anti-derivative properties that very much resemble derivative properties. So we can break this over some so we can factor out coefficients. So the derivative of 9.8 times the integral of t dt minus 25 times the integral of dt. In which case the anti-derivative of t is going to be a t squared over 2. The anti-derivative of 1 is just going to be t. And then we also have this some unknown constant c that we have to deal with in just a second. Let's see if we can simplify this a little bit better. If you take 9.8 divided by 2, you're going to get 4.9. So negative 4.9 t squared minus 25 t plus a constant. We get so. So this gives us the function, but we also know the initial value. The initial height that is s at 0, that is when no time has elapsed, the initial height is going to be 1000. So if we plug 0 into our function, we're going to negative 4.9 times 0 squared minus 25 times 0 plus c, right? We see that this simplifies just to be c and so the initial value is going to be 1000. So with that in hand, we now have our position function s of t equals negative 4.9 t squared minus 25 t plus 1000. That's our position function. Then we have to figure out what would the position after 10 seconds. So we're trying to figure out what is s of 10 in that situation. So we're going to get negative 4.9 times 10 squared, which is 100 minus 25 times 10 plus 1000 times it by 10 is pretty nice because it just kind of moves the decimal place around. So if you times by 100, that's going to move the decimal place by two positions. We're going to negative 490 times the 25 by 10 adds a zero at the end. So we get negative 250 and then we add that to 100. And of course, if you take, excuse me, 1000, sorry, 1000 take away 250, that's 750. And then we have to take 490 away from 750. That's going to end up with 260. And so we see that after 10 seconds, the ball will be 260 meters above the ground.