Question Type 12: Formulating exponential models given descriptions Exercises

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

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.

Write down an exponential model for a bacteria population that grows at a rate of 2 per second, with an initial population of 8 and a baseline amount of 3.

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

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.

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

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 $P(t)$.

Light intensity in water decreases exponentially with depth. The intensity approaches a background level of 5 lux and drops by 40% per meter from an initial intensity of 100 lux above background. Write $I(d)$ as a function of depth $d$ in meters.

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

A machine bought for $50{,}000 depreciates continuously at 12% per year toward a salvage value of $5{,}000. Write the value model $V(t)$.