 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 $4$ 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 $t=0$)?

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