Solve the problem.
-The function $P ( t ) = 8 ( 1.0134 ) ^ { t }$ represents the population (in millions) of a country $t$ years after January 1, 2000. Write an equivalent function using base $e$; that is, write a function of the $P ( t ) = P _ { 0 } e ^ { r t }$. Round $r$ to 5 decimal places. Also, determine the population for the year 2000 .
A) $P ( t ) = 8 e ^ { 0.01331 t }$; population in 2000: 8 million
B) $P ( t ) = 8 e ^ { 0.01331 t }$; population in $2000 : 8.1$ million
C) $P ( t ) = 2.1 e ^ { 0.10649 t }$; population in 2000: $2.3$ million
D) $P ( t ) = 2.1 e ^ { 0.10649 t }$; population in $2000 : 2.1$ million

Question

Quizplus · Accepted Answer

The given function is $P(t) = 8(1.0134)^t$. To convert this to a function with base $e$, we use the formula $P(t) = P_0e^{rt}$, where $e$ is the base of the natural logarithm. We need to find $r$ such that $e^r = 1.0134$. Taking the natural logarithm of both sides, we get $r = \ln(1.0134)$, which approximately equals $0.01331$ when rounded to five decimal places. Therefore, the equivalent function is $P(t) = 8e^{0.01331t}$. For the year 2000, or $t = 0$, $P(0) = 8(1.0134)^0 = 8$, so the population in 2000 was 8 million.

Solve the Problem P(t)=8(1.0134)tP ( t ) = 8 ( 1.0134 ) ^ { t }P(t)=8(1.0134)t

Solve the Problem $P ( t ) = 8 ( 1.0134 ) ^ { t }$