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# Quadratic word problem: ball

CCSS.Math:

## Video transcript

a ball is shot into the air from the edge of a building 50 feet above the ground its initial velocity is 20 feet per second 20 feet per second the equation H I'm guessing H is 4 height is equal to negative 16t squared plus 20 T plus 50 can be used to model the height of the ball after T seconds and I think in this problem they just want us to accept this formula although we do derive formulas like this and show the why it works for this type of problem in the Khan Academy physics playlist but for here we'll just go with the flow on this example so they give us the equation that can be used to model the height of the ball after T seconds and then say about how long does it take for the ball to hit the ground so if this is the height if this is the height the ground is when the height is equal to zero so hitting the ground means this literally means that H is equal to zero so we need to figure out what times at which times does H equals zero so we're really solving the equation zero is equal to negative 16t squared plus twenty T plus plus 50 and if you want to simplify this a little bit let's see everything here is divisible at least by two and let's divide everything by negative two just so that we can get rid of this negative leading coefficient so you divide the left-hand side by negative two you still get a zero negative sixteen divided by negative two is eight so eight T squared twenty divided by negative two is negative 10 minus 10 T 50 divided by negative two is minus 25 and so we have eight T squared minus 10 t minus 25 is equal to zero or if you're comfortable with this on the left hand side we can put it on the left hand side we could just say this is equal to zero and now we solve and we could complete the square here or we can just apply the quadratic formula which is derived from completing the square and we have this in standard form we know that this is our a this is our a this right over here is our B and this over here is our C and the quadratic formula tells us that the roots and in this case it's in terms of the variable T are going to be equal to negative B plus or minus the square root of B squared minus 4ac all of that over all of that over 2a so if we apply it we get T is equal to negative B B is negative 10 so negative negative 10 is going to be positive 10 plus or minus the square root of negative 10 squared well that's just positive 100 minus 4 times a which is 8 times C which is negative 25 negative 25 and all of that over 2a a is 8 so 2 times 8 is 16 and this over here we have a let's see how if we can simplify this a little bit the negative sign negative times a negative these are going to be positive 4 times 25 is 100 times 8 is 800 so all that simplifies to 800 and we have a hundred plus 800 under the radical sign so let me so this is equal to 10 plus or minus the square root of 900 all of that over 16 and this is equal to 10 plus or minus square root of 900 is 30 over 16 and so we get time is equal to 10 plus 30 over 16 is 40 over 16 which is the same thing if we divide the numerator and the denominator by 4 to simplify it as 10 over or actually even better divide it by 8 that's 5 over 2 so that's one solution if we add the third if we subtract the 30 we'd get 10 minus 30 so or T is equal to 10 minus 30 which is negative 20 over 16 divide the numerator and the denominator by 4 you get negative 5 over 4 now we had to remember this was we're trying to find a time and so a time at least in what this problem that we're dealing with we should only think about positive times we want to figure out how much time has taken Pat how long does it take for the ball to hit the ground we don't want to go back in time so we don't want our negative answer right here so we only want to think about our positive answer and so this tells us that the only route that should work is 5 halves and we assumed that this is in seconds so this is in 5 halves seconds I wouldn't worry too much about the physics here I think they really just want us to apply the quadratic formula to this modeling situation the physics we go into a lot more depth give you the kind of the conceptual understanding on the on our physics playlist but let's verify that we definitely are at a height of zero at five 1/2 seconds or if T is equal to five halves this expression right over here does give us H is equal to zero so we have let's try it out we have negative 16 times five halves squared times five halves squared plus 20 times five halves plus 50 this is going to be equal to this needs to be equal to zero this needs to be equal to zero so this is negative 16 times 25 over 4 plus let's see if we divide 20 by 2 we get 10 if we divide 2 by 2 we get 1 so 10 times 5 is going to be 50 plus 50 this needs to be equal to zero negative 16 divided by 4 is negative 4 4 divided by 4 is 1 so you have negative 4 times 25 you have which is 100 plus 50 I'm sorry negative 4 times 25 is negative 100 plus 50 plus 50 again is equal to zero and so we have 100 negative 100 plus another hundred well that's definitely going to be equal to 0 we get zero equals zero and it all checks out we hit the ground after five and a half I've halves seconds or another way to think about it is 2.5 seconds 5 hat T is equal to 5 half seconds or 2.5 seconds