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

Name: Problem 28, Chapter 2: Calculus with Applications
Uploaded: 2020-03-14T18:27:08-08:00
Duration: 4 min 49 s
Channel: Robert Daugherty
Description: Decrease in Bacteria When an antibiotic is introduced into a culture of 20,000 bacteria, the number of bacteria decreases exponentially. After 12 hours, there are only 5000 bacteria left. (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 20,000 bacteria, the number of bacteria decreases exponentially. After 12 hours, there are only 5000 bacteria left. (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 20,000 bacteria, the number of bacteria decreases exponentially. After 12 hours, there are only 5000 bacteria left.
(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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