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Trig word problem: modeling annual temperature

Video transcript

the hottest day of the year in santiago chile on average is january 7th when the average high temperature is 29 degrees celsius the coolest day of the year has an average high temperature 14 degrees celsius use a trigonometric function to model the temperature in santiago chile using 365 days as the length of a year remember that january 7th is the summer in santiago how many days after january 7th is the first spring day when the temperature reaches 20 degrees celsius so let's do this in first parts in two parts so first let's let's use a trigonometric let's try to figure out a trigonometric function that models the temperature in santiago chile so we'll have temperature as a function of days where days are the number where days are the number of days after january 7th and once we have that trigonometric function to model that then we can answer the second part i guess the essential question which is how many days after january 7th is the first spring day when the temperature reaches 20 degrees celsius so to think about this let's graph it let's graph it and it should become pretty apparent why they are suggesting that we use a trigonometric function to model this because our seasonal variations they're cyclical they go up and down and actually if you look at the average temperature for any city over the course of the year it really does look like a trigonometric function so let's so let's this axis right over here this is the passage of the days let's do d for days and that's going to be in days after january 7th so this right over here would be january 7th and the horse and the vertical axis this is the horizontal the vertical axis is going to be in terms of celsius degrees so this let's see the high is 29 and i could write 29 degrees celsius and then or that the highest average day and then if this is 0 then 14 which is the lowest average day 14 degrees celsius and so our temperature will vary between these two extremes our temperature is going to vary but the highest average day which they already told us is january 7th we get to 29 degrees celsius and then the coldest day of the year on average you get to you get to an average high of 14 degrees celsius so it looks like this we're talking about average highs on a given day and the reason why it's why a trigonometric function is a good idea is because it's cyclical so this is january 7th if you go 365 days in the future you're back at three you're back at january 7th so this is the if the average high temperature is 29 degrees celsius on that day the average high temperature is going to be 29 degrees celsius on that day now we're using a trigonometric function so we're going to hit our low point exactly halfway in between so we're going to hit our low point exactly halfway in between something right like that and so our function is going to look like this our function so let me see let me just draw the low point right over there and this is a high point it's a high point right over there that looks pretty good and then i have the high point right over here and then i just need to connect them and there you go i've drawn one period of our trigonometric function and our period is 365 days if we go 365 days later we're at the same point in the cycle we are at the same point in the year we're at the same point in the year so what i want you to do right now is given what i've just drawn try to model this right so this is this right over here is let's write this this is capital t as a function of d try to figure out an expression for capital t as a function of d remember it's going to be some trigonometric function so i'm assuming you've given a go at it and you might say well this could look like a cosine curve maybe it could be a sine curve which one should i use you could actually use either one but i always like to go with the simpler one they just think about well if this was if these were angles either actually degrees or radians which trigonometric function starts at your maximum point well cosine of 0 is 1. the cosine starts at your maximum point sine of 0 is 0. so i'm going to use cosine here i'm going to use a cosine function so temperature as a function of days is going to be some amplitude times our cosine function and we're going to have some argument to our cosine function and then i'm probably going to have to shift it so let's think about how we would do that well what's the midline here well the midline's the halfway point between our high and our low so our middle point if we were to visualize it looks just like so that is our midline right over there and what value is this well what's the average of 29 and 14. 29 plus 14 is 43 divided by 2 is 21.5 degrees celsius so that's our midline so essentially we've shifted up our function by we've shifted up our function by that amount if we just had a regular cosine function our midline would be at zero but now we're at 21.5 degrees celsius so i'll just write plus 21.5 that's how much we've shifted it up now what's the amplitude well our amplitude is how much we diverge from the midline so over here we're 7.5 above the midline so that's plus 7.5 here we're 7.5 below the midline so minus 7.5 so our amplitude is 7.5 our maximum va the maximum amount we go away from the midline is 7.5 so that's our amplitude and now let's think about our argument to the cosine function right over here so we're going to have our it's going to be a function of the days and what do we want when when 365 days have gone by we want our cosine we want this entire argument to do be 2 pi so when d is 365 we want this whole thing about to evaluate to 2 pi so we could put 2 pi over 365 in here and you probably you might remember your formulas i always forget them that's why i always try to reason through them again the formula is oh you want 2 pi divided by your period and all the rest but i just like to think okay look after one period which is 365 days i want the whole argument over here to be 2 pi i wanted i want to go around the unit circle once and so if this is 365 if this is 2 pi over 365 when you multiply by 365 your argument here is going to be 2 pi so just like that we've done the first part of this question