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From compound interest to bubonic plague, things that grow or spread really fast are often modeled by exponential functions. Learn about these powerful functions (pun intended?).
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?
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\(pH\) 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 \(100a?\)
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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?
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If you invest \(1200\) dollars in savings account that pays \(4\) percents compound interest, then how long does it take to double your money?
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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 \(t=0\))?
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