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CCSS.Math: , ,

Nik and ELISA are racing up a wall Alyssa's height on the wall is given by the equation a is equal to 1/3 T plus 5 so that looks like I was like their wall climbers of some kind where a is Alyssa's height and feet after climbing 40 seconds Nick started racing at the same time as ELISA and is also climbing at a constant speed his height is shown in the following table so this is T and seconds time in seconds this is height and feet who started out higher ELISA or Nick so to figure out their starting position we just need to figure out what was their height at time equals zero that's when this whole race started for elicit is pretty straightforward when time is equal to 0 you have one-third times zero plus five well that's just going to be five feet so ELISA starting position is at five feet when time is equal to zero now let's think about Nick's height at time equals zero and there's a couple of ways that we can go about doing this one is is just to back up to kind of go backwards on this chart so let me show you what I'm talking about so if this is time and this is let's say n for Nick's height because we have a for Alyssa's height so let me make a little table here we already know that at time six seconds or after six seconds he's six feet in the air along the wall at time eight he is seven feet in the air or along the wall at a time ten he has gotten to a height of eight feet so what's happening here every time every time two seconds goes by every time two seconds goes by he increases in one foot he increases one foot you have another two seconds he increases in height by one foot so you could go backwards if we take away two seconds to four seconds he will decrease in height by one foot if we go back another two seconds he will decrease in height by another one foot the reason why we can say this is because we know he's climbing at a constant speed so if we decrease by another two seconds to our starting time then we know that his he would have been one foot lower so he would have been at three feet so just like that we now know what time equals euro Knicks height is three feet in the air so ELISA started out higher than Nick so this right over here would be the correct answer now the other way to do it is set up an equation just like we had for ELISA and substitute for time equals zero and the way to do that is to recognize that Nick's height as a function of time is also going to be a linear equation because they're both climbing they're both climbing at constant speed or we know that Nick is climbing at a constant speed so Nick's height as a function of time is going to look like Nick's height is going to be some slope some rate of change essentially his height per his height per second at times time plus his initial plus his initial position so how can we solve for M the slope and his initial position well the slope is just his rate of change of height so it's literally how much does his height change per unit time so M right over here M is just going to be for a unit time for a change in time how much is his height changing and his height is we used the letter n so we already know that when time increases by 2 when time increases by to his height increases by his height increases by one foot by one foot so we know that M is equal to one-half he increases one-half feet per second and you see that there because it takes some two seconds to go one foot so we can fill in M here so we know now that n n is equal to one-half one-half T plus B now to solve for B you could just substitute one of these points all of these points must satisfy this equation right over here so we could use the point six so if we put a 6 in here so when time is six we know that n is 6 so you have 6 is equal to 1/2 times 6 plus B or you get 6 is equal to 3 plus B subtract 3 from both sides you get B is equal to three so there you have it you get Nick's equation as a fun Ickx height as a function of time Nick as a function of time is going to be equal to one-half T plus three so now we have an equation just like Alyssa's and we can say well when time is equal to zero he's at a height of three which is lower than Alyssa's initial height