Exponential Functions
The population of a certain culture of bacteria at time \(t\) is given by \(P(t) = P_0 \times 10^{yt} \), where \(P_0\) is the initial amount of bacteria, \(y\) is the growth constant, and \(t\) is the time (in seconds). A scientist measures that the number of bacteria at time \(t=4\) is \(10^ {56}\) times the number of bacteria at time \(t=0\). Given this information, what is the growth constant \(y\) of this group of bacteria?
\(pH\) measures the concentration of hydrogen ions \(H^+\) in a solution. Let \(pH=m\) if the number of gram ions of \(H^+\) in a \(1\)-liter solution is \(10^{-m}\).
Then the ratio of the number of gram ions of \(H^+\) for \(pH=6.5\) and the number of gram ions of \(H^+\) for \(pH=7.9\) can be expressed as \(10^a.\) What is the value of \(100a?\)
The brightness of light is reduced by half when it penetrates a glass screen with \(11\) cm width. If the brightness of light has been reduced to \(12.50\) % of the initial brightness, how many glass screens with the same width did the light penetrate?
If you invest \(1200\) dollars in savings account that pays \(4\) percent compound interest, then how long does it take to double your money? (Assume the answers are in years and the compounding occurs once a year.)
The number of particles in a given location after time \(t\) is modeled by the equation \(N(t) = N_0e^{-\frac{t}{21}}\). At what time \(t\) will the number of particles be equal to \(\frac{1}{e^3}\) of the initial number (when \(t=0\))?