Question Type 4: Finding parameters of exponential models using the initial condition Exercises

Given $f(t)=a\cdot2^t$ and $f(0)=10$, find $a$.

Given $N(t)=a e^{0.5t}$ and $N(0)=200$, find $a$.

An asset depreciates according to $V(t)=a(0.9)^t$ and is worth $20,000 at $t=0$. Find $a$.

Species introduced to a lake follow the model $P(t)=a\cdot3^{0.1t}$. If the initial population is 50, find $a$.

A bacteria culture doubles every hour. If the initial population is 1,200, write the model $N(t)=a\cdot2^t$ and find $a$.

A radioactive substance decays with a half-life of 5 hours according to $P(t)=a\,(0.5)^t$. If the initial mass is 80 g, find $a$.

An investment grows continuously at 3% per year described by $V(t)=a e^{0.03t}$. If its value at $t=0$ is $1,000, find $a$.

Given $D(t)=a\cdot b^t$, if $D(0)=5$ and $D(3)=320$, find $a$ and $b$.

Demand over time is modeled by $D(t)=a b^t$, with $D(0)=100$ and $D(4)=200$. Find $a$ and $b$.

Given the exponential growth model $M(t)=a e^{kt}$, if $M(0)=30$ and $M(1)=45$, find $a$ and $k$.

Given $E(t)=a e^{kt}$, with $E(0)=15$ and $E(5)=30$, find $a$ and $k$.

A radioactive decay is given by $R(t)=a e^{kt}$. If the half-life is 5 years and $R(0)=250\text{ g}$, find $k$ and $a$.