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An epidemic follows the curve


P = 500 / 1+20,000e^(-0.549t)


; where t is in years. How fast is the epidemic growing after 10 years? (Round your answer to two significant digits.)

Verified Answer :

Edufirst

P(t) = 500 / [1+20,000e^(-0.549t) ]

How fast means rate of change of the function, and that is its derivative.

Then, find the derivative of P(t), P '(t), using the chain rule:

P '(t) = - 500 / [ 1 + 20,000e^(-0.549t)]^2 ] 20,000(-0.549)e^(-0.549)t

P '(t) = 5,490,000 e^ (-0.549t) / [1 + 20,000e^-0,549t)]^2

P'(t) = 5,490,000 e^(0.549t) / [ e^(0.549t) + 20,000]^2

Now replace t by 10 .

P'(10) =5,490,000 * e^(5.49) / [ e^(5.49) + 20,000]^2 = 3.25 years

Answer: 3.3 years



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