Solution - Episode 2; The "Heat Wave" in Winnipeg

Hey folks, it's Lawrence here again, and I'll be solving the Transformation question.

During episode two, the Guardian asks these Questions:

An Average daily maximum temperature in Winnipeg follows a sinusoidal pattern with a lowest value of 8.4 on April 7th, and a highest value of 30.9 on August 4

A) Sketch the graph and describe the variation with 2 equations
- a sine and a cosine function.

B) What is the expected Maximum temperature for July 13th?

C) Algebraically determine how many days will have an expected maximum of 23.0 or higher

A) Sketch the graph and describe the variation with 2 equations
- a sine and a cosine function

So we'll start with the graph since we need to find the equation to solve B and C anyways. This function is described by the equation t(d) = A (sin or cos) [B (d - C)] + D ; Where t is Temperature, d is Days (time), A is the Amplitude, B is the Period, C is the Phase Shift and D is the Sinusoidal Axis.

So to start off, we'll find D using the maximum and minimum value to find the Sinusoidal Axis. Add 30.9 and 8.4, and then divide it by two to get the middle.

D = 30.9 + 8.4 / 2 = 19.65.

Great, now we have our D value! From here, we can find the A value. We do so by taking the maximum value and subtracting the sinusoidal axis so that we find the amplitude of the wave.

A = 30.9 - 19.65 = 11.25

Ok! So we now have the A and D values! Next, we'll find the B value, which is the period. B is defined by 2π/period, so we'll find that by having to add up all the days.

First, we'll have to start at April the 7th, because that's when our lowest value occurs earliest in the year. So we find the sum of the days up until August the 4th. So we go;

Apr .7 + May 31 + Jun. 30 + Jul. 31 + Aug. 4 = 103

Because that's only a fraction of the period, we have to find the next 3 points where the temperature hits the sinusoidal axis, the maximum and back to the sinusoidal axis. So from 103, we have to add the month's we haven't yet. It's not like they didn't exist so we have to count them in the period. So we'll have to add the 23 remaining days in April, the 31 days in March, the 28 days in February, and the 31 days in January. So...

103 + 27 + 31 + 28 + 31 = 216

That's only half of the period, so we'll have to multiply it by two.

216 * 2 = 432.

We now have the B value which is 2π/432. So far we have

A = 11.25 B = 2π/432 D = 19.65

Next, we'll have to find C if we want to fully graph this thing. So because our lowest value occurs on April the 7th, we can find the Phase Shift for the Cosine Function of this Equation. So basically, we have to add up the days from the first day in the year, until April the 7th. Therefore...

31 + 28 + 31 + 7 = 97.

For the cosine function, the Phase Shift is 97, but to find it for the Sine Function, we'll have to add it to the first quarter of the Event. To do that, we'll take the Period, which is 432, divide it by two to get half, which is 216, and the divide it by two again to get a quarter, which is 108. We will then add the 97 to 108 to find the Phase Shift of the Sine Function.

108 + 97 = 205.

We must also add 97 to the half point and the third quarter of the event if we want to properly draw the graph out since it THERE IS A PHASE SHIFT.

216 + 97 = 313
324 + 97 = 421

GREAT! Now that we have all our "tools" we can now write out the sine and cosine function for this event, and draw it as well.

A = 11.25, B = 2π/432, C = 97 or 205, D = 19.65

SINE FUNCTION: t(d) = 11.25sin[2π/432(d - 205)] + 19.65

COSINE FUNCTION: t(d) = 11.25cos[2π/432(d - 97)] + 19.65

And this is what the graph would look like (Click the image)

B) What is the expected Maximum temperature for July 13th?

To solve this one is pretty straight forward. You want to find how many days you have for 'd' to plug into your function, and then just go from there. So we have to add all the days up until July the 13th.

31+28+31+30+31+30+13 = 194

So now we plug it into the Function and also, we'll use the sine function because we won't have to deal with negative values. t(194) = 11.25sin[2π/432(194 - 205)] + 19.65.
There's nothing special really here. You just plug it into your handy dandy calculator and you come up to an answer of 19.6185, which we can round to 19.62 degrees Celsius.

C) Algebraically determine how many days will have an expected maximum of 23.0 or higher.

Now, when we solve this one, there's a little trick in how you treat the function so that you don't end up messing yourself up. Okay, so we want to find how many days will have an expected maximum of 23 degrees Celsius or higher.

First, we'll let θ = [2π/432(d - 205)], just so that it's easier to deal with. So the function ends up looking a little like this: 23.0 = 11.25sinθ + 19.65. From there we'll subtract 19.65 from both sides. We then get 3.35 = 11.25sinθ and we'll reduce 11.25 by dividing it from both sides. We then get 0.2977 = sinθ. We'll use the Arc sine function on our calculators to isolate θ.

After that, we get 0.3022 = θ. Place back in [2π/432(d - 205)] and we'll then have 0.3022 = [2π/432(d - 205)].

Multiply the reciprocal of 2π/432 from both sides and we'll have 20.7890 = d - 205. Add 205 to both sides to isolate d and find our answer in Days.

get a final answer of 225.7890. We can round that to 226 days or even change the decimals into hours and minutes.

And that's it folks!

Rence ~ Out