Consider the exponential growth model

MATHEMATICS · Subjective · AP · Calculus AB · Differential Equations · Differential Equations

Problem

P(t)=P_0 e^{k t}

, where

P_0

is the initial amount and

k

is a constant. Explain, with complete algebraic reasoning, how measurements of the population

P(t_1)

at time

t_1

and

P(t_2)

at a later time

t_2

allow determination of the constant

k

without requiring knowledge of

P_0

. Derive an explicit formula for

k

in terms of the two measured values and the time interval

(t_2-t_1)