Question

Scientists have discovered a vaccine for a debilitating disease. This year there were 570,000 reported new cases of the disease in the United States. Research indicates, that with the new vaccination, new cases will decrease by two-thirds each year.

How many new cases do you estimate in 6 years?

Answer 1

Answer: There are approximately 853827 new cases in 6 years.

Explanation:

Since we have given that

Initial population = 570000

Rate at which population decreases is given by

`(2)/(3)`

Now,

First year =570000

Second year is given by

`570000* ((1)/(3))`

Third year is given by

`570000((1)/(3))^2`

so, there is common ratio ,

it becomes geometric progression, as there is exponential decline.

so,

`570000,570000* (1)/(3),570000*( (1)/(3))^2,......,570000* ((1)/(3))^6`

a=570000

common ratio is given by

`r=(a_2)/(a_1)=(1)/(3)`

number of terms = 6

Sum of terms will be given by

`S_n=(a(1-r^n))/((1-r))`

We'll put this value in this formula,

`S_6=(570000(1-((1)/(3))^6)/((1-(1)/(3)))\\\\=853827.16`

So, there are approximately 853827 new cases in 6 years.