You are put in charge of terraforming and designing a tropical paradise planet.

Question

You are put in charge of terraforming and designing a tropical paradise planet. The ideal vacation spots on this planet are a group of large islands south of the equator. Create a sine or cosine equation modeling its yearly temperature cycle near the islands.

-Choose a (integer) number of earth days in which a year will elapse on this planet, and use this to form the period.

-Choose a distinct high and low temperature (both > 0 deg Celsius) throughout the year and use this to form the amplitude and vertical shift.

-Suppose the year starts on the coldest day of the year. Account for this with a phase shift or another property.

-Write the completed equation model for the temperature, where time is measured in days.

-Sketch a graph of the temperature over one year. (You can use graphing software to assist/check.)

Explanation / Answer

Rolling a single die

1) probability of rolling divisors of 6 :

Since its a single die, the possible outcomes are 1,2,3,4,5,6. All have equal probability(1/6) since its a fair die

Out of these divisors of 6 are 1,2,3,6. So P(divisors of 6) = 1/6*1/6*1/6*1/6 = 1/1296 = 0.0008

2) probability of rolling a multiple of 1: Since all(1,2,3,4,5,6) are multiples of 6 = 1/6*1/6*1/6*1/6*1/6*1/6= 1/46656 = 0.00002

3) probability of rolling an even number : There are 3 even numbers between 1-6 i.e. 2,4,6

Hence probability of rolling an even number = 1/6*1/6*1/6 = 1/216 =0.0046

4) List of all possible outcomes of rolling a single die ={1,2,3,4,5,6}

5) probability of rolling factors of 3 : Factors of 3 are 1,3

Hence probability of rolling factors of 3 = 1/6*1/6 = 1/36 = 0.0278

6) probability of rolling a 3 or smaller : 3 or smaller are 1,2,3. Hence the probability = 1/6*1/6*1/6 = 1/216 = 0.0046

7) probability of rolling a prime number: Prime numbers between 1-6 are 2,3,5 hence probability = 1/6*1/6*1/6=1/216=0.0046

8) probability of rolling factors of 4 : Factors of 4 are 1,2,4 hence the probability = 1/6*1/6*1/6 = 1/216 =0.0046

9) probability of rolling divisors of 30 : Divisors of 30 are 1,2,3,5,6 = 1/6*1/6*1/6*1/6*1/6 = 1/7776 = 0.0001

10) probability of rolling factors of 24: Factors of 24 are 1,2,3,4,6 = 1/6*1/6*1/6*1/6*1/6=1/7776=0.0001

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