Decrease in Bacteria When an antibiotic is introduced into a...

Decrease in Bacteria When an antibiotic is introduced into a culture of $50,000$ bacteria, the number of bacteria decreases exponentially. After 9 hours, there are only $20,000$ bacteria. a. Write an exponential equation to express the growth function $y$ in terms of time $t$ in hours. b. In how many hours will half the number of bacteria remain?

Decrease in Bacteria When an antibiotic is introduced into a culture of $50,000$ bacteria, the number of bacteria decreases exponentially. After 9 hours, there are only $20,000$ bacteria.
a. Write an exponential equation to express the growth function $y$ in terms of time $t$ in hours.
b. In how many hours will half the number of bacteria remain?

Key Concepts

Exponential Decay

Exponential decay is a process in which a quantity decreases at a rate proportional to its current value. This is modeled by an equation where the rate of change is constant relative to the current amount, typically using a negative exponent to indicate a decrease over time, which is fundamental in describing processes such as radioactive decay, cooling, and the effect of antibiotics on bacteria.

Exponential Function

An exponential function describes a relationship in which a quantity changes by a constant multiplicative factor over equal intervals of time. The standard form, y = y? · e^(kt), uses an initial value y? and a constant k, where a negative k represents decay. This function is crucial for modeling scenarios where change accelerates or decays rapidly.

Decay Constant

The decay constant is a parameter in an exponential decay equation that determines the rate at which the decay occurs. It is derived from known data points by solving the exponential model for k and is essential for predicting how quickly the quantity decreases over time.

Half-life

Half-life is the time required for a decaying quantity to reduce to half of its original amount. In the context of exponential decay, it provides a measure of the speed of the process and is calculated by setting the exponential decay equation to one-half of the initial value and solving for time using logarithmic methods.

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