A species of duckweed (floating freshwater aquatic plant) that can be found in f

Question

A species of duckweed (floating freshwater aquatic plant) that can be found in freshwater ponds and slow-moving streams has been studied by scientists in field and laboratory settings. It has been found that its biomass W(t) (measured in grams) as a function of time t (days) can be described by an exponential function in the form W(t) = ce^kt , where c and k are constants.

A. In day 0, the biomass of duckweed was 0.0394 grams and the rate of change was 0.0047. Use this information to find c and k.

B. Find the duckweed biomass in day 12.

C. Find the instantaneous rate of change of the duckweed biomass in day 12.

Explanation / Answer

(A)Given function

W(t) = cekt .......................(1)

when t=0, W=0.0394, dw/dt=0.0047

replacing t=0, W=0.0394    in (1)

0.0394= c e0

c=0.0394

dw/dt= ck ekt

0.0047= 0.0394 k e0

0.0047/0.0394=k

k=0.1192

(B)W(t) = cekt replacing c and k

W(t) =0.0394 e.01192t

when t=12,

W(t)= 0.0394 e.01192 * 12

=0.1647 grams

(C) instantaneous rate of change of the duckweed biomass=dw/dt= 0.0394 * 0.1192 e0.1192 t

replacing t=12

dw/dt=0.0394 * 0.1192 e0.1192 * 6

=0.00960

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