Under certain circumstances, a rumor spreads according to the equation p(t)=1/1+ae−kt where p(t) is the proportion of the population that knows the rumor at time t, and a and k are positive constants. (a) find limt→[infinity]p(t). (b) find the rate of speed of the rumor. (c) how long will it take for 80% of the population to hear the rumor? here, take a=10,k=0.5 with t measured in hours.

(a)

(b)

(c) 80% of the population will have heard the rumor in about 7.4 hrs.

Step-by-step explanation:

We know that a rumor spreads according to the equation

where is the proportion of the population that knows the rumor at time , and and are positive constants.

(a) To find you must:

Use this fact,

Apply this identity, to find

(b) To find the rate of speed of the rumor you must find the derivative

(c) To find the time that will take for 80% of the population to hear the rumor, you must substitute a = 10, k = 0.5, and p(t) = 0.8 into and solve for t

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step-by-step explanation: