Question

Suppose the function P(t) = 437.6(1.031)t is used to model the population of an organism in a specific region after t years. When will the number of organisms be 1000? (Round your answer to three decimal places.)

Answer 1

Answer:  27.071  years.

Step-by-step explanation:

The given function : $P(t) = 437.6(1.031)^t$ is used to model the population of an organism in a specific region after t years.

To find : t , when P(t)=1000

Substitute  P(t)=1000 in the given function , we get

$1000 = 437.6(1.031)^t\\\\\Rightarrow\ \dfrac{1000}{437.6}=(1.031)^t\\\\\Rightarrow\ 2.2852=(1.031)^t$

Taking natural log on both sides , we get

$\ln(2.2852)=\ln ((1.031)^t)\\\\\Rightarrow\ \ln(2.2852)=t(\ln (1.031))\\\\\Rightarrow\ t=\dfrac{ \ln(2.2852}{\ln(1.031)}\\\\\Rightarrow\ t=\dfrac{0.8264535}{0.0305292}\\\\\Rightarrow\ t=27.070918989\approx27.071$

Hence, The number of organisms will be 1000 after t= 27.071  years.