Question

Each of the following situations can be modelled using an exponential function. Indicate which situations require a value of \(c >1\) (growth) and which require a value of \(0 < c < 1\) (decay). Explain your choices. a) Bacteria in a Petri dish double their number every hour. b) The half-life of the radioactive isotope actinium-225 is 10 days. c) As light passes through every \(1-\mathrm{m}\) depth of water in a pond, the amount of light available decreases by 20\%. d) The population of an insect colony triples every day.

Short answer

a) Growth (\(c > 1\))b) Decay (\(0 < c < 1\))c) Decay (\(0 < c < 1\))d) Growth (\(c > 1\))

Step-by-step solution

Understand the concept of exponential growth and decay

Exponential functions can describe situations where quantities grow or decay over time. In the function form, it is generally written as \[f(t) = a \times c^t\] where \(a\) is the initial amount, \(c\) is the growth (if \(c > 1\)) or decay (if \(0 < c < 1\)) factor, and \(t\) is the time.

Analyze the first situation

a) Bacteria in a Petri dish double their number every hour. Since the number of bacteria is doubling, the growth factor is greater than 1. This is an example of exponential growth where \(c > 1\).

Analyze the second situation

b) The half-life of the radioactive isotope actinium-225 is 10 days. Since the amount of isotope is halved every 10 days, this indicates a decay. Hence, the value of \(c\) should be between 0 and 1, indicating \(0 < c < 1\).

Analyze the third situation

c) As light passes through every 1-meter depth of water in a pond, the amount of light available decreases by 20%. Since the amount of light decreases by a percentage (20%), this is a decay situation where \(c\) is less than 1, specifically, \(0.8\) in this case (\(0 < c < 1\)).

Analyze the fourth situation

d) The population of an insect colony triples every day. Since the population triples, it grows exponentially with a factor greater than 1. Therefore, this is an example of exponential growth where \(c > 1\).

Key concepts

exponential growth

Exponential growth describes a situation where a quantity increases by a consistent percentage over equal intervals of time. For example, in the problem, the bacteria in a Petri dish double every hour. This means the number of bacteria increases exponentially, doubling each hour.
Exponential growth can be modeled with the function \(f(t) = a \times c^t\), where \( a \) is the initial quantity, \( c \) is the growth factor (greater than 1), and \( t \) represents time.
Common examples of exponential growth include population growth, where a population multiplies over time, and investments accruing interest.

exponential decay

Exponential decay is the opposite of exponential growth. It describes a situation where a quantity decreases by a consistent percentage over equal intervals of time.
In the aforementioned exercise, the half-life of actinium-225 and the decrease in light through water are both examples of exponential decay.
The general formula for exponential decay is \(f(t) = a \times c^t\), where \(c\) (between 0 and 1) represents the decay factor, causing the quantity to get smaller over time.
Common occurrences of exponential decay include radioactive decay and depreciation of assets.

half-life

Half-life is a specific case of exponential decay. It describes the time needed for a quantity to reduce to half its initial value.
In the solution, actinium-225 has a half-life of 10 days, meaning every 10 days, half of the remaining radioactive isotope decays.
The formula for half-life can be expressed as \(N(t) = N_0 \times \frac{1}{2}^{(\frac{t}{t_{1/2}})}\), where \(N(t)\) is the amount remaining at time \(t\), \(N_0\) is the initial amount, and \(t_{1/2}\) is the half-life period.
Understanding half-life helps in fields like chemistry, physics, and environmental science to predict the decay of substances.

population growth

Population growth is a common example of exponential growth in nature. It describes how populations (like bacteria, insects, or even human populations) increase over time, often by reproducing at constant rates.
In the provided exercise, the population of an insect colony triples every day, showing exponential growth.
The general model for population growth is \(P(t) = P_0 \times c^t\), where \(P_0\) is the initial population and \(c\) is the growth factor.
Factors influencing population growth include birth rates, death rates, immigration, and emigration. Understanding these concepts is valuable in fields like ecology, biology, and demography.