# 4.1: Growth and Decay

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- William F. Trench
- Andrew G. Cowles Distinguished Professor Emeritus (Mathematics) at Trinity University

This section begins with a discussion of exponential growth and decay, which you have probably already seen in calculus. We consider applications to radioactive decay, carbon dating, and compound interest. We also consider more complicated problems where the rate of change of a quantity is in part proportional to the magnitude of the quantity, but is also influenced by other other factors for example, a radioactive substance is manufactured at a certain rate, but decays at a rate proportional to its mass, or a saver makes regular deposits in a savings account that draws compound interest.

Since the applications in this section deal with functions of time, we’ll denote the independent variable by \(t\). If \(Q\) is a function of \(t\), \(Q'\) will denote the derivative of \(Q\) with respect to \(t\); thus,

\[Q'={dQ\over dt}.\nonumber \]

## Exponential Growth and Decay

One of the most common mathematical models for a physical process is the *exponential model*, where it is assumed that the rate of change of a quantity \(Q\) is proportional to \(Q\); thus

\[\label{eq:4.1.1} Q'=aQ,\]

where \(a\) is the constant of proportionality.

From Example 2.1.3, the general solution of Equation \ref{eq:4.1.1} is

\[Q=ce^{at}\nonumber \]

and the solution of the initial value problem

\[Q'=aQ, \quad Q(t_0)=Q_0\nonumber \]

is

\[\label{eq:4.1.2} Q=Q_0e^{a(t-t_0)}.\]

Since the solutions of \(Q'=aQ\) are exponential functions, we say that a quantity \(Q\) that satisfies this equation *grows exponentially* if \(a > 0\), or *decays exponentially* if \(a < 0\) (Figure 4.1.1
).

## Radioactive Decay

Experimental evidence shows that radioactive material decays at a rate proportional to the mass of the material present. According to this model the mass \(Q(t)\) of a radioactive material present at time \(t\) satisfies Equation \ref{eq:4.1.1}, where \(a\) is a negative constant whose value for any given material must be determined by experimental observation. For simplicity, we’ll replace the negative constant \(a\) by \(-k\), where \(k\) is a positive number that we’ll call the *decay constant* of the material. Thus, Equation \ref{eq:4.1.1} becomes

\[Q'=-kQ.\nonumber \]

If the mass of the material present at \(t=t_0\) is \(Q_0\), the mass present at time \(t\) is the solution of

\[Q'=-kQ,\quad Q(t_0)=Q_0.\nonumber \]

From Equation \ref{eq:4.1.2} with \(a=-k\), the solution of this initial value problem is

\[\label{eq:4.1.3} Q=Q_0e^{-k(t-t_0)}.\]

The *half–life* \(\tau\) of a radioactive material is defined to be the time required for half of its mass to decay; that is, if \(Q(t_0)=Q_0\), then

\[\label{eq:4.1.4} Q(\tau+t_0)={Q_0\over 2}.\]

From Equation \ref{eq:4.1.3} with \(t=\tau+t_0\), Equation \ref{eq:4.1.4} is equivalent to\[Q_0e^{-k\tau}={Q_0\over 2},\nonumber \]

so

\[e^{-k\tau}={1\over 2}.\nonumber \]

Taking logarithms yields

\[-k\tau=\ln{1\over 2}=-\ln2,\nonumber \]

so the half-life is

\[\label{eq:4.1.5} \tau={1\over k}\ln2.\]

(Figure 4.1.2 ). The half-life is independent of \(t_0\) and \(Q_0\), since it is determined by the properties of material, not by the amount of the material present at any particular time.

##### Example 4.1.1

A radioactive substance has a half-life of 1620 years.

- If its mass is now 4 g (grams), how much will be left 810 years from now?
- Find the time \(t_1\) when 1.5 g of the substance remain.

**Solution a**

From Equation \ref{eq:4.1.3} with \(t_0=0\) and \(Q_0=4\),

\[\label{eq:4.1.6} Q=4e^{-kt},\]

where we determine \(k\) from Equation \ref{eq:4.1.5}, with \(\tau\)= 1620 years:

\[k={\ln2\over\tau}={\ln2\over 1620}. \nonumber\]

Substituting this in Equation \ref{eq:4.1.6} yields

\[\label{eq:4.1.7} Q=4e^{-(t\ln2)/1620}.\]

Therefore the mass left after 810 years will be

\[\begin{array}{rl} Q(810) &=4e^{-(810\ln2)/1620}=4e^{-(\ln2)/2} \\ &=2\sqrt{2} \mbox{ g}. \end{array}\nonumber \]

**Solution b**

Setting \(t=t_1\) in Equation \ref{eq:4.1.7} and requiring that \(Q(t_1)=1.5\) yields

\[{3\over2}=4e^{(-t_1\ln2)/1620}. \nonumber\]

Dividing by 4 and taking logarithms yields

\[\ln{3\over8}=-{t_1\ln2\over1620}. \nonumber\]

Since \(\ln3/8=-\ln8/3\),

\[t_1=1620{\ln8/3\over\ln2}\approx 2292.4\;\mbox{ years}. \nonumber\]

## Interest Compounded Continuously

Suppose we deposit an amount of money \(Q_0\) in an interest-bearing account and make no further deposits or withdrawals for \(t\) years, during which the account bears interest at a constant annual rate \(r\). To calculate the value of the account at the end of \(t\) years, we need one more piece of information: how the interest is added to the account, or—as the bankers say—how it is *compounded*. If the interest is compounded annually, the value of the account is multiplied by \(1+r\) at the end of each year. This means that after \(t\) years the value of the account is

\[Q(t)=Q_0(1+r)^t. \nonumber \]

If interest is compounded semiannually, the value of the account is multiplied by \((1+r/2)\) every 6 months. Since this occurs twice annually, the value of the account after \(t\) years is

\[Q(t)=Q_0\left(1+{r\over 2}\right)^{2t}. \nonumber \]

In general, if interest is compounded \(n\) times per year, the value of the account is multiplied \(n\) times per year by \((1+r/n)\); therefore, the value of the account after \(t\) years is

\[\label{eq:4.1.8} Q(t)=Q_0\left(1+{r\over n}\right)^{nt}. \]

Thus, increasing the frequency of compounding increases the value of the account after a fixed period of time. Table 4.1.1 shows the effect of increasing the number of compoundings over \(t=5\) years on an initial deposit of \(Q_0=100\) (dollars), at an annual interest rate of 6%.

\(n\) (number of compoundings per year) |
\($100\left(1+\frac{.06}{n} \right)^{5n}\) (value in dollars after 5 years) |
---|---|

1 | $133.82 |

2 | $134.39 |

4 | $134.68 |

8 | $134.83 |

364 | $134.98 |

You can see from Table 4.1.1
that the value of the account after 5 years is an increasing function of \(n\). Now suppose the maximum allowable rate of interest on savings accounts is restricted by law, but the time intervals between successive compoundings isn’t; then competing banks can attract savers by compounding often. The ultimate step in this direction is to *compound continuously*, by which we mean that \(n\to\infty\) in Equation \ref{eq:4.1.8}. Since we know from calculus that

\[\lim_{n\to\infty} \left(1+{r\over n}\right)^n=e^r, \nonumber\]

this yields

\[\begin{array}{rl} Q(t) & =\lim_{n\to\infty} Q_0\left(1+{r\over n}\right)^{nt}=Q_0 \left[ \lim_{n\to\infty} \left(1+{r\over n}\right)^n\right]^t \\[12pt] &=Q_0e^{rt}. \end{array} \nonumber \]

Observe that \(Q=Q_0e^{rt}\) is the solution of the initial value problem

\[Q'=rQ, \quad Q(0)=Q_0; \nonumber\]

that is, with continuous compounding the value of the account grows exponentially.

##### Example 4.1.2

If $150 is deposited in a bank that pays \(5{1\over2}\)% annual interest compounded continuously, the value of the account after \(t\) years is

\[Q(t)=150e^{.055t} \nonumber\]

dollars. (Note that it is necessary to write the interest rate as a decimal; thus, \(r=.055\).) Therefore, after \(t=10\) years the value of the account is

\[Q(10)=150e^{.55} \approx \$259.99. \nonumber\]

##### Example 4.1.3

We wish to accumulate $10,000 in 10 years by making a single deposit in a savings account bearing \(5{1\over2}\)% annual interest compounded continuously. How much must we deposit in the account?

**Solution**

The value of the account at time \(t\) is

\[\label{eq:4.1.9} Q(t)=Q_0e^{.055t}.\]

Since we want \(Q(10)\) to be $10,000, the initial deposit \(Q_0\) must satisfy the equation

\[\label{eq:4.1.10} 10000=Q_0e^{.55},\]

obtained by setting \(t=10\) and \(Q(10)=10000\) in Equation \ref{eq:4.1.9}. Solving Equation \ref{eq:4.1.10} for \(Q_0\) yields

\[Q_0=10000e^{-.55} \approx \$5769.50.\nonumber \]

## Mixed Growth and Decay

##### Example 4.1.4

A radioactive substance with decay constant \(k\) is produced at a constant rate of \(a\) units of mass per unit time.

- Assuming that \(Q(0)=Q_0\), find the mass \(Q(t)\) of the substance present at time \(t\).
- Find \(\lim_{t\to\infty} Q(t)\).

**Solution a: **

Here

\[Q'=\mbox{ rate of increase of } Q - \mbox{ rate of decrease of } Q.\nonumber \]

The rate of increase is the constant \(a\). Since \(Q\) is radioactive with decay constant \(k\), the rate of decrease is \(kQ\). Therefore

\[Q'=a-kQ.\nonumber \]

This is a linear first order differential equation. Rewriting it and imposing the initial condition shows that \(Q\) is the solution of the initial value problem

\[\label{eq:4.1.11} Q'+kQ=a, \quad Q(0)=Q_0.\]

Since \(e^{-kt}\) is a solution of the complementary equation, the solutions of Equation \ref{eq:4.1.11} are of the form \(Q=ue^{-kt}\), where \(u'e^{-kt}=a\), so \(u'=ae^{kt}\). Hence,

\[u={a\over k}e^{kt}+c\nonumber \]

and

\[Q=ue^{-kt}={a\over k}+ce^{-kt}. \nonumber \]

Since \(Q(0)=Q_0\), setting \(t=0\) here yields

\[Q_0={a\over k}+c \quad \text{or} \quad c=Q_0-{a\over k}. \nonumber \]

Therefore

\[\label{eq:4.1.12} Q={a\over k}+\left(Q_0-{a\over k}\right)e^{-kt}.\]

b. Since \(k > 0\), \(\lim_{t\to\infty} e^{-kt}=0\), so from Equation \ref{eq:4.1.12}

\[\lim_{t\to\infty} Q(t)={a\over k}. \nonumber \]

This limit depends only on \(a\) and \(k\), and not on \(Q_0\). We say that \(a/k\) is the *steady state* value of \(Q\). From Equation \ref{eq:4.1.12} we also see that \(Q\) approaches its steady state value from above if \(Q_0 > a/k\), or from below if \(Q_0 < a/k\). If \(Q_0=a/k\), then \(Q\) remains constant (Figure 4.1.3
).

## Carbon Dating

The fact that \(Q\) approaches a steady state value in the situation discussed in Example 4 underlies the method of *carbon dating*, devised by the American chemist and Nobel Prize Winner *W.S. Libby*.

Carbon 12 is stable, but carbon-14, which is produced by cosmic bombardment of nitrogen in the upper atmosphere, is radioactive with a half-life of about 5570 years. Libby assumed that the quantity of carbon-12 in the atmosphere has been constant throughout time, and that the quantity of radioactive carbon-14 achieved its steady state value long ago as a result of its creation and decomposition over millions of years. These assumptions led Libby to conclude that the ratio of carbon-14 to carbon-12 has been nearly constant for a long time. This constant, which we denote by \(R\), has been determined experimentally.

Living cells absorb both carbon-12 and carbon-14 in the proportion in which they are present in the environment. Therefore the ratio of carbon-14 to carbon-12 in a living cell is always \(R\). However, when the cell dies it ceases to absorb carbon, and the ratio of carbon-14 to carbon-12 decreases exponentially as the radioactive carbon-14 decays. This is the basis for the method of carbon dating, as illustrated in the next example.

##### Example 4.1.5

An archaeologist investigating the site of an ancient village finds a burial ground where the amount of carbon-14 present in individual remains is between 42 and 44% of the amount present in live individuals. Estimate the age of the village and the length of time for which it survived.

**Solution**

Let \(Q=Q(t)\) be the quantity of carbon-14 in an individual set of remains \(t\) years after death, and let \(Q_0\) be the quantity that would be present in live individuals. Since carbon-14 decays exponentially with half-life 5570 years, its decay constant is

\[k={\ln2\over 5570}. \nonumber\]

Therefore

\[Q=Q_0e^{-t(\ln2)/5570}\nonumber\]

if we choose our time scale so that \(t_0=0\) is the time of death. If we know the present value of \(Q\) we can solve this equation for \(t\), the number of years since death occurred. This yields

\[t=-5570 {\ln Q/Q_0\over\ln2}.\nonumber\]

It is given that \(Q=.42Q_0\) in the remains of individuals who died first. Therefore these deaths occurred about

\[t_1=-5570 {\ln.42\over\ln2} \approx 6971 \nonumber\]

years ago. For the most recent deaths, \(Q=.44 Q_0\); hence, these deaths occurred about

\[t_2=-5570 {\ln.44\over\ln2} \approx 6597 \nonumber\]

years ago. Therefore it is reasonable to conclude that the village was founded about 7000 years ago, and lasted for about 400 years.

## A Savings Program

##### Example 4.1.6 :

A person opens a savings account with an initial deposit of $1000 and subsequently deposits $50 per week. Find the value \(Q(t)\) of the account at time \(t > 0\), assuming that the bank pays 6% interest compounded continuously.

**Solution**

Observe that \(Q\) isn’t continuous, since there are 52 discrete deposits per year of $50 each. To construct a mathematical model for this problem in the form of a differential equation, we make the simplifying assumption that the deposits are made continuously at a rate of $2600 per year. This is essential, since solutions of differential equations are continuous functions. With this assumption, \(Q\) increases continuously at the rate

\[Q'=2600+0.06 Q \nonumber\]

and therefore \(Q\) satisfies the differential equation

\[\label{eq:4.1.13} Q'-.06Q=2600.\]

(Of course, we must recognize that the solution of this equation is an approximation to the true value of \(Q\) at any given time. We’ll discuss this further below.) Since \(e^{.06t}\) is a solution of the complementary equation, the solutions of Equation \ref{eq:4.1.13} are of the form \(Q=ue^{.06t}\), where \(u'e^{.06t}=2600\). Hence, \(u'=2600e^{-.06t}\),

\[u=- {2600\over.06}e^{-0.06t}+c \nonumber\]

and

\[\label{eq:4.1.14} Q=ue^{.06t}=-{2600\over.06}+ce^{.06t}.\]

Setting \(t=0\) and \(Q=1000\) here yields

\[c=1000+{2600\over 0.06}, \nonumber\]

and substituting this into Equation \ref{eq:4.1.14} yields

\[\label{eq:4.1.15} Q=1000e^{.06t}+{2600\over.06}(e^{.06t}-1) \]

where the first term is the value due to the initial deposit and the second is due to the subsequent weekly deposits.

Mathematical models must be tested for validity by comparing predictions based on them with the actual outcome of experiments. Example 6 is unusual in that we can compute the exact value of the account at any specified time and compare it with the approximate value predicted by Equation \ref{eq:4.1.15} (See *Exercise 4.1.21*). The following table gives a comparison for a ten year period. Each exact answer corresponds to the time of the year-end deposit, and each year is assumed to have exactly 52 weeks.

Year |
Approximate Value of \(Q\) (Example 4.1.6
) |
Exact Value of \(P\) (Exercise 4.1.21) |
Error \(Q-P\) |
Percentage Error \((Q-P)/P\) |
---|---|---|---|---|

\(1\) | \($3741.42\) | \($3739.87\) | \($1.55\) | \(.0413%\) |

\(2\) | \(6652.36\) | \(6649.17\) | \(3.19\) | \(.0479\) |

\(3\) | \(9743.30\) | \(9738.37\) | \(4.93\) | \(.0506\) |

\(4\) | \(13,025.38\) | \(13,018.60\) | \(6.78\) | \(.0521\) |

\(5\) | \(16,510.41\) | \(16,501.66\) | \(8.75\) | \(.0530\) |

\(6\) | \(20,210.94\) | \(20,200.11\) | \(10.83\) | \(.0536\) |

\(7\) | \(24,140.30\) | \(24,127.25\) | \(13.05\) | \(.0541\) |

\(8\) | \(28,312.63\) | \(28,297.23\) | \(15.40\) | \(.0544\) |

\(9\) | \(32,742.97\) | \(32,725.07\) | \(17.90\) | \(.0547\) |

\(10\) | \(37,447.27\) | \(37,426.72\) | \(20.55\) | \(.0549\) |