A bacteria culture starts with 500 bacteria and doubles in size every hour. How many bacteria will there be after 3 hours?

Carbon-14 has a half-life of 5730 years. A fossil bone contains 20% of its original Carbon-14. Estimate the age of the fossil.

An initial investment of $10,000 grows at an annual interest rate of 5% compounded continuously. How long will it take for the investment to double in value?

A metal bar at 100°C is placed in a room at 20°C. After 10 minutes, the temperature of the bar is 80°C. Find the time it takes to cool to 40°C.

A cold drink at 5°C is brought into a room at 25°C. After 15 minutes, its temperature rises to 12°C. What will its temperature be after 30 minutes?

A tank initially contains 100 liters of pure water. Brine containing 0.2 kg of salt per liter enters the tank at a rate of 5 liters/minute. The well-mixed solution leaves the tank at the same rate. Find the amount of salt in the tank after 20 minutes.

A tank initially contains 500 gallons of brine with 50 lbs of dissolved salt. Brine containing 2 lbs of salt per gallon flows in at 5 gal/min. The well-mixed solution flows out at 3 gal/min. Find the amount of salt in the tank after $t$ minutes.

A 12-volt battery is connected to a series circuit with a resistor of 10 ohms and an inductor of 2 henries. The initial current is zero. Find the current $i(t)$.

An RC series circuit has a resistance of 200 ohms, a capacitance of $10^{-4}$ farads, and an applied voltage of 100V. If the initial charge on the capacitor is zero, find the charge $q(t)$.

A skydiver of mass $m=80$ kg jumps from an airplane. The air resistance is proportional to velocity, with a constant $k = 16$ kg/s. If the initial velocity is zero, find the velocity function $v(t)$ and the terminal velocity. Use $g = 9.8 \text{ m/s}^2$.

Find the orthogonal trajectories for the family of parabolas $y = cx^2$.

Find the orthogonal trajectories of the family of concentric circles $x^2 + y^2 = c^2$.