Linear function word problems — Harder example

min Lee's house is located 1.4 miles from her school when she walks home from school it takes her an average of 24 minutes assuming that min Lee walks at a constant rate constant rain we could figure that out because we know how far she walks in a certain amount of time which of the following functions best models min Lee's distance from home D in miles so we want distance in miles if she has walked a total of T minutes so T is going to be in minute so all are you interested me minutes and miles and that's good because they gave us things in terms of minutes and miles if this was in seconds or hours we would have to do some conversion so let's just think about her rate her constant rate that they're talking about the rate that min Lee walks well she covers 1.4 miles she covers 1.4 miles in 24 minutes 24 minutes well this isn't a pretty fraction with the decimal in the numerator so let's let's multiply the numerator and the denominator by 10 so that we get rid of this decimal so that's going to be equal to 14 over 240 miles per minute and then we can further simplify that the numerator denominator both divisible by two so we this is going to be let's see fourteen divided by two is seven 240 divided by two over 120 miles per minute miles per minute so we were able to figure out Millie's constant rate now we need to figure out D we need to figure out D so we have to be very careful D is min Lee's distance from home from home she's leaving from school and it's her distance from home remember her home is 1.4 miles from her school so there's a bunch of ways that we can tackle it but maybe the easiest one is well what's the what's her distance if you wanted to say distance from school distance from school distance from school that's just going to be her rate times T times how much time has passed by as when time is zero she's gonna be at her school as time increases in minutes she's going to get further and further away from her school but her distance from home her distance from home is going to be one point four miles minus the distance from school so let me just write D because this is what we care about D which is her distance in miles from home distance in miles from home that's going to be that's going to be one point 4 minus distance from school - distance from school and if that doesn't make sense just think about it if this is her home right over here I'll write H for home if this is her school right here s for school we know that this distance right over here is one point four miles now if she has walked if she has walked say I don't let me this in another color if she has walked I don't know point four miles if this is zero point four miles right over here then her distance from home is gonna be one point four miles - that it's going to be it's going to be this distance right over here so distance from home is going to be one point four minus the distance from school and what's that going to be well that's just going to be one point four minus the rate times the time well what's the rate seven one twenty its miles per minute and we got the unit's right and so this is a D is going to be equal to one point four minus seven over 128 T and if we look at the choices that's going to be this first choice over here and it's what I look at the other choices and to think about well how could we've ruled them out fairly quickly well this one has one point for the distance between the two places - 24 T well this isn't the rate right over here that's that's how long it takes her to walk it's not the rate so you could rule that one out this one is one point four - the reciprocal of the rate so that's a strange answer and this one this one makes it look like she's getting further and further from home as time gets bigger notice this this one right over here it does correctly say at time zero she's going to be one point four miles from her home which is accurate because at times you or she's going to be at her school but then after 1 minute after 2 minutes after 3 minutes she's going to get in based on this model further and further away from her home so this is this would be a case where she's walking away from home from her school so you would rule that one out as well