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Algebra

# Exponential Functions - Growth and Decay

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 $$4$$ percents compound interest, then how long does it take to double your money?

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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