Question

The normal average daily temperature in degrees Fahrenheit for a city is given by the equation shown below, where (t) is the time in days, with (t = 1) corresponding to January 1. (T = 60 - 24 cosleft(frac{2pi (t - 32)}{365} ight)) (a) Find the expected date of the warmest day. The warmest day is between July 10 and July 11. The warmest day is between July 4 and July 5. The warmest day is between August 12 and August 13. The warmest day is between August 2 and August 3. (b) Find the expected date of the coldest day. The coldest day is January 5. The coldest day is December 25. The coldest day is March 10. The coldest day is February 1.

          The normal average daily temperature in degrees Fahrenheit for a city is given by the equation shown below, where (t) is the time in days, with (t = 1) corresponding to January 1.
(T = 60 - 24 cosleft(frac{2pi (t - 32)}{365}
ight))
(a) Find the expected date of the warmest day.
The warmest day is between July 10 and July 11.
The warmest day is between July 4 and July 5.
The warmest day is between August 12 and August 13.
The warmest day is between August 2 and August 3.
(b) Find the expected date of the coldest day.
The coldest day is January 5.
The coldest day is December 25.
The coldest day is March 10.
The coldest day is February 1.

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$The normal average daily temperature in degrees Fahrenheit for a city is given by the equation shown below, where (t) is the time in days, with (t = 1) corresponding to January 1. (T = 60 - 24 cosleft(frac2pi (t - 32)365 ight)) (a) Find the expected date of the warmest day. The warmest day is between July 10 and July 11. The warmest day is between July 4 and July 5. The warmest day is between August 12 and August 13. The warmest day is between August 2 and August 3. (b) Find the expected date of the coldest day. The coldest day is January 5. The coldest day is December 25. The coldest day is March 10. The coldest day is February 1.$

Added by Esperanza S.

Introductory and Intermediate Algebra for College Students 4th

Robert Blitzer 4th Edition

Instant Answer

Solved by Expert Darshan Maheshwari

12/24/2021

Step 1

This happens when the argument of the cosine function, 2t - 32, equals 0 or any multiple of 2π. Solving 2t - 32 = 0 gives t = 16 days, which corresponds to January 17. However, we need to consider the period of the cosine function, which is 2π. Show more…

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Transcript

00:01 This question, the normal average daily temperature in degrees fahrenheit for cit is given by the equation shown below where t is timed days.

00:08 But t equal to 1 corresponding to january 1.

00:12 So we need to find the expected date of the warmest day.

00:15 So warm as day would mean that the temperature is the highest.

00:17 And when will the temperature be highest? when cosine will be minus 1? why i'm saying minus 1? because minus 1 is this minimum value.

00:26 And where it is minus, this minus will get multiplied with minus.

00:30 24 because there's a minus over here and it will generate a plus sign so that would be 60 plus 24 which is 84 degrees which is the warmest day so that is possible only when cost of 2 pi times t minus 32 over 365 is equal to minus 1 so this is possible when 2 pi times t minus 32 over 365 will be pi.

00:59 So if this is pi, then these two are canceled.

01:01 Then the value of t minus 32 using a cross multiplication will be 365 over 2.

01:06 So the value of t is coming out as let me just grab my calculator.

01:11 That's coming as 365 over 2 plus 32.

01:15 So that is 214 .5.

01:20 That is the value of t.

01:21 It means that 214th day of the month corresponds to the warmest.

01:29 So that is one solution, by the way.

01:32 There can be other solution as well since cosine is minus 1, not only when it is pi, it is also minus 1 when it is 3 pi as well.

01:44 So another solution can be when this becomes 3.

01:48 So here we have 3.

01:49 So here we'll have 3.

01:50 So another value of p would be three times 365 over 2 plus 32.

01:57 I hope you understand why i'm saying 3 because this will now i will write as 3 pi.

02:01 So 3 will keep on coming over here.

02:03 So this will be another value of time.

02:07 So this will be 365 times 3 over 2 plus 32, which is actually exceeding the value of 500, the value of the year, which is, it's coming as 579...

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