One model for the way diseases die out when properly treated assumes that the ra

Question

One model for the way diseases die out when properly treated assumes that the rate dy / dt at which the number of infected people changes is proportional to the number y. The number of people cured is proportional to the number that have the disease. Suppose that in any given year, the number of cases of a disease is reduced by 25%. There are 10,000 cases today. How long will it take to reduce the number of cases to 1000? How long will it take to eradicate the disease, that is, reduce the number of cases to less than 1? Is x

Explanation / Answer

let equation is

y = y0 e^kt

now to start with begining..

at t = 0 we have 10,000 cases

thus

y = 10,000 e^kt

at t = 1 no of cases will be 75% of its present value

thus or 7500

7500 = 10,000 e^k

e^k =0.75

k = ln(0.75) < 0

y = 10,000 e^(ln(0.75)t

a) to become y = 1000

1000 = 10,000 e^(ln(0.75)t)

0.1 = e^(ln(0.75)t

ln(0.1) = ln(0.75) t

t = ln(0.1) / ln(0.75) = 8.003922779651

t = 8 years.

b) to become y = 1

1 = 10,000 e^ln(0.75)t

1/10000 = e^ln(0.75) t

0.0001 = e^ln(0.75)t

ln(0.0001) = ln(0.75)t

t = 32.015691118604

t = 32.01 years.






e)
pi^e = 22.45915771836104
e^pi = 23.140692632779263

e^pi > pi^e

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