Question

In the town of Gossip, every rumor starts with one person telling that rumor to three other people. Every 15 minutes each person who has heard the rumor tells it to three more people, and then doesn't tell anyone else. How long will it be before 100 people hear the rumor? How about 1,000? 10,000?

Answer 1

Answer:

(a)100 people will hear the rumor before 90 minutes

(b)1000 people will hear the rumor before 120 minutes

(c)10000 people will hear the rumor before 150 minutes

Step-by-step explanation:

The rumor starts with one person telling that rumor to three other people.

Each person who has heard the rumor tells it to three more people.

Since the next term is a ratio of the previous term, this is a geometric progression where:

The first term a=1

Common ratio =3

The nth term of a Geometric sequence, $U_n=ar^{n-1}$

• We want to find out how long before 100 people hear the rumor

$U_n=ar^{n-1}\\100=1 X 3^{n-1}\\100=3^{n-1}$

$3^{n-1}=100\\3^4<3^{n-1}=100<3^5\\3^4<3^{n-1}<3^5\\\4<n-1<5\\5<n<6$

Since n is greater than 5 but less than 6, we know 100 people will hear the rumor after 75 minutes but before 90 minutes.

• We want to find out how long before 1000 people hear the rumor

$U_n=ar^{n-1}\\1000=1 X 3^{n-1}\\1000=3^{n-1}$

$3^{n-1}=1000\\3^6<3^{n-1}=1000<3^7\\3^6<3^{n-1}<3^7\\6<n-1<7\\7<n<8$

Since n is greater than 7 but less than 8, we know 1000 people will hear the rumor after 105 minutes but before 120 minutes.

• We want to find out how long before 10000 people hear the rumor

$U_n=ar^{n-1}\\10000=1 X 3^{n-1}\\10000=3^{n-1}$

$3^{n-1}=10000\\3^8<3^{n-1}=10000<3^9\\3^8<3^{n-1}<3^9\\8<n-1<9\\9<n<10$

Since n is greater than 9 but less than 10, we know 10000 people will hear the rumor after 135 minutes but before 150 minutes.

REMARK(FOR ILLUSTRATION)

We know n is between 5 and 6. To get the exact time, If we change to logarithm form

$3^{n-1}=100$

$Log_{3}100=n-1\\\frac{Log 100}{Log 3} =n-1\\4.19=n-1\\n=4.19+1=5.19$

Since each cycle is 15 Minutes,

In 15 X 5.19 minute=77.85 minutes = 1 hour 18 Minutes

In approximately 1 hour 18 Minutes, 100 people will hear the rumor.