# Trig word problem: modeling annual temperature

## Video transcript

Voiceover:The hottest day of
the year in Santiago, Chile on average, is January
seventh, when the average high temperature is 29 degrees Celsius. The coolest day of the year has an average high temperature of 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 seventh
is the summer in Santiago. How many days after January
seventh is the first spring day when the temperature
reaches 20 degrees Celsius? So let's do this in two parts. First, let's try to figure
out a trigonometric function that models the temperature
in Santiago, Chile. We'll have temperature
as a function of days, where days are the number of
days after January seventh. 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 seventh
is the first spring day when the temperature
reaches 20 degrees Celsius?" To think about this 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. 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. 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 seventh. So this right over here
would be January seventh. And the vertical axis, this
is the horizontal axis. The vertical axis is going to be in terms of Celsius degrees. The high is 29 and I could
write 29 degrees Celsius. The highest average day. If this is zero then 14, which
is the lowest average day. 14 degrees Celsius. So our temperature will vary
between these two extremes. The highest average day,
which they already told us, is January seventh we get
to 29 degrees Celsius. And then the coldest day
of the year, on average, 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 a
trigonometric function is a good idea is because it's cyclical. If this is January
seventh, if you go 365 days in the future, you're
back at January seventh. 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. So our function is
going to look like this. Our function, let me see. I'm going to draw the low point right over there and this is a high point. That's a high point right over there. That looks pretty good. 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 through 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 right over here, let's write this as T as a function of d. Try to figure out an expression
for T as a function of d and 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 looks 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. Just think about well
if these were angles, either actual degrees or radians, which trigonometric function
starts at your maximum point? Well cosine of zero is one. The cosine starts at your maximum point. Sine of zero is zero, so I'm
going to use cosine here. I'm going to use a cosine function. So, temperature as a function of days. There's 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 mid line here? The mid line is the halfway point between our high and our low. So our midpoint, if we were to visualize it, looks just like so. That is our mid line right over there. And what value is this? Well what's the average of 29 and 14? 29 plus 14 is 43 divided by
two is 21.5 degrees Celsius. So that's our mid line
so essentially we've shifted up our function by that amount. If we just had a regular
cosine function our mid line would be at zero, but now
we're at 21.5 degrees Celsius. 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 mid line. Over here we're 7.5 above the
mid line so that's plus 7.5. Here we're 7.5 below the
mid line, so minus 7.5. So our amplitude is
7.5, the maximum amount we go away from the mid line is 7.5. So that's our amplitude. And now let's think about our argument to the cosine function right over here. It's going to be a function of the days. And what do we want? When 365 days have gone by, we want this entire argument to be two pi. So when d is 365, we want this whole thing to evaluate to two pi. We could put two pi over 365 in here. You might remember your
formulas, I always forget them that's why I always try to
reason through them again. The formulas, you want two
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 two pi. "I want to go around the unit
circle once and so if this "is two pi over 365, when
you multiply it by 365 "your argument here is
going to be two pi." Just like that we've done the
first part of this question. We have modeled the
average high temperature in Santiago as a function of
days after January seventh. In the next video we'll
answer this second question. I encourage you to do
it ahead of time before watching that next video
and I'll give you one clue. Make sure you pay attention to the fact that they're saying the first spring day.