1. Suppose a culture of bacteria begins with \(5000\) cells and dies by \(30\%\) each year (i.e. yearly compounding). Which equation represents this situation.

2. The number of mosquitoes at the beginning of the summer was \(4000\). The population of mosquitoes is expected to grow at a rate of \(25\%\) a month (i.e. monthly compounding). How many mosquitoes will there be after \(4\) months?

3. The number of bacteria in a culture is growing at a rate of \(3000 {\rm{e}}^{2t/5}\) per unit of time \(t\). At \(t = 0\), the number of bacteria present was \(7500\). Find the number present at \(t = 5\).

4. The formula for exponential growth of the quantity \(N(t)\) is

5. The volume of substance \(A\), measured in cubic centimeters, increases according to the exponential growth formula \(\frac{\textrm{d} A}{\textrm{d} t} = 0.3 A\), where \(t\) is measured in hours. The volume of another quantity \(B\), also measured in cubic centimeters, increases at a constant rate of \(1\) cm\(^3\)/h according to the linear growth model \(\frac{\textrm{d} B}{\textrm{d} t} = 1\). At \(t=0\), substance \(A\) has a volume \(A(0) = 3\) and substance \(B\) has a volume \(B(0) = 5\). At which time will both substances have the same volume?

6. Suppose a population of bears grows according to a logistic rate, i.e. \[ \frac{\textrm{d} P}{\textrm{d} t} \, \, = \, \, 2P - 0.01 P^2 \, \, = \, \, 2P \left( 1 - 0.005 P \right) \, , \] where \(P(t)\) is the number of bears at time \(t\) in years. Which of the following statements is true?
I) If \(P > 200\), the population of bears is increasing.
II) If \(P > 200\), the population of bears is decreasing.
III) \(\lim_{t \to \infty} P(t) = 200\).

7. As in question {\bf{6}} assume again that a population of bears grows according to the logistic rate \(\dot{P} = 2P - 0.01 P^2\). At which population is its growth rate the greatest?

8. A fossilized leaf contains \(18\%\) of its normal amount of carbon 14. How old is the fossil (to the neares year)? Use \(5600\) years as the half-life of carbon 14.

9. The loudness of sound is measured in units called decibels (dB). These units are measured by first assigning an intensity \(I_0\) to a very soft sound that is called the threshold sound. The sound to be measured is assigned an intensity \(I\) and the decibel rating \(d\) of this sound is found using \[ d \, \, = \, \, 10 \cdot \lg\left( \frac{I}{I_0} \right) \, . \] The threshold sound audible to the average person is \(1.0 \cdot 10^{-12} W/m^2\) (watts per square meter). One considers the following sound level classification \[ \begin{array}{c c c c} {\text{moderate}} & {\text{loud}} & {\text{very}} {\text{loud}} & {\text{deafening}} \\ \hline 45-69 \, {\text{dB}} & 70-89 \, {\text{dB}} & 90-109 \, {\text{dB}} & > 110 \, {\text{dB}} \end{array} \] How would a sound with intensity \(6.3 \cdot 10^{-3}\) \(W/m\)\(^2\) be classified?

10. A coffee is sitting on your desk cooling. It cools according to the function \(T = 70 \cdot (0.80)^x + 20\), where \(x\) is the time in minutes and \(T\) is the temperature in degree Celsius. What was the initial temperature of the coffee?