Question Type 12: Formulating exponential models given descriptions Exercises

A certain isotope has a half-life of 10 days. If you start with 200 mg, write an exponential model $I(t)$ for the mass remaining after $t$ days.

A small mammal population stabilizes at a baseline of 50 individuals. On top of this, it grows by 1.5% per year. If the total population is 200 at $t=0$, write a model $P(t)$.

A machine bought for $50\,000$ depreciates continuously at $12\%$ per year toward a salvage value of $5000$. Write the value model $V(t)$.

An investment of $1000 is compounded continuously at an annual rate of 4%. Write the value $V(t)$ after $t$ years.

A deposit of $5000$ dollars is invested at an annual interest rate of $6\%$, compounded quarterly. Write a model $A(t)$ for the account balance after $t$ years.

Express $I(d)$ as a function of depth $d$ in metres.

A radioactive substance decays by 5% each year, starting with 100 g. Write an exponential model for its mass $M(t)$ after $t$ years.

An object at $90^\circ\mathrm{C}$ cools toward an ambient temperature of $25^\circ\mathrm{C}$ with constant $k=0.1 \, \mathrm{h}^{-1}$. Write the temperature model $T(t)$ after $t$ hours. [3]

Determine an exponential model for a bacteria population that has an initial population of 8 and a baseline amount of 3. The population is modeled by an exponential function with base 2.

Carbon-14 has a half-life of 5730 years. If an artifact contains 100 units initially, write the model $C(t)$ for the remaining units after $t$ years.

After an injection, the concentration of a drug in the bloodstream decays continuously toward a baseline level of $2\text{ mg/L}$ at a rate constant of $0.3$ per hour. The concentration above baseline is $50\text{ mg/L}$ initially. Write the concentration model $C(t)$.