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

A bacteria culture doubles every hour. If the initial population is $1\ 200$, find the value of $a$ for the model $N(t) = a \cdot 2^t$ and state the resulting model.

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 the value of $a$.

The mass of a radioactive substance is given by $R(t) = a e^{kt}$, where $t$ is the time in years and $R(t)$ is the mass in grams. Given that the half-life of the substance is 5 years and the initial mass is $R(0) = 250 \text{ g}$, find the values of $k$ and $a$.

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

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

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

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

A radioactive substance decays according to the model $P(t)=a\,(0.5)^t$, where $t$ is the time in hours. If the initial mass is 80 g, find the value of $a$.

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

An investment grows continuously at 3% per year described by $V(t)=a e^{0.03t}$. If its value at $t=0$ is 1000, find $a$.

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