A ship embarked on a long voyage. At the start of the voyage, there were 500 ant

Question

A ship embarked on a long voyage. At the start of the voyage, there were 500 ants in the cargo hold of the ship. One week into the voyage, there were 800 ants. Suppose the population of ants is an exponential function of time. (a) How long did it take the population to double? (Round your answer to the nearest 0.01.) (b) How long did it take the population to triple? (Round your answer to the nearest 0.01.) (c) When were there be 10,000 ants on board? (Round your answer to the nearest 0.01.) (d) There also was an exponentially-growing population of anteaters on board. At the start of the voyage there were 17 anteaters, and the population of anteaters doubled every 2.8 weeks. How long into the voyage were there 200 ants per anteater? (Round your answer to the nearest 0.01.)

Explanation / Answer

1)

general exponential equation is y=a*bt

y(0)=500, y(1)=800

a*b0=500 =>a=500

500*b1=800

=>b =1.6

(a)

y=500*1.6t

population doubles =>500*1.6t=500*2

=>t=ln2/ln1.6

=>t=1.47 weeks

b)

population triples =>500*1.6t=500*3

=>t=ln3/ln1.6

=>t=2.34 weeks

c)

population=10000

500*1.6t=10000

=>t=ln20/ln1.6

=>t=6.37 weeks

(d)

for anteaters  exponential equation be z=c*dt

c=17

population doubles every 2.8 weeks

17*d2.8=17*2

=>d=21/2.8

=>equation for ant eaters is z=17*2t/2.8

200ants per anteater

=>y/z=200

=>y=200z

=>500*1.6t=200*17*2t/2.8

=>(1.6/21/2.8)t=200*17/500

=>1.249t=6.8

=>t=ln6.8/ln1.249

=>t=8.62 weeks

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