Population of a city doubles in $50$ years. In how many years will it triple?

Question

The population of a city doubles in $50$ years. In how many years will it triple, under the assumption that the rate of increase is proportional to the number of inhabitants?

This was a pretty trivial question. But, the thing is that when I tried solving this as:

We have, the population of a city becoming double in $50$ years. This means, that the population increases by $100$ percent in $50$ years. This hints at the fact that the population was increasing at the rate of $2$ percent per year. So, we consider the initial population as $P$.

Let the number of years in which it becomes triple is $y$ years. Then, we have the following equation holding true,

$$\left(P+\frac{2Py}{100}\right)=3P\implies y=100\text{years}.$$

So, I thought, the population will become thrice in $100$ years.

But here, comes the problem. The answer given suggests the population becomes thrice in $50\log_23$ years. I don't understand the how is it so? Have I done some error/or forgotten to take something into consideration?

Answer 1

Well, in this context, the city's population grows exponentially, not linearly (which means as time goes on, the rate of growth increases/decreases in a curve and doesn't stay at a single value). You'd be right if it does grow linearly – but it doesn't!

Now, we can think of the population as a function of time, something like $n\cdot2^t$ where $t$ is the time and $n$ is the initial population.

Let's say that $t$ is measured in a half-century, or 50 years (so that $t=2$ represents 100 years, etc). For each 50 years that pass by ($t=1$), the population will increase from $n$ to $n\cdot2^1$, hence doubling the population. So, this way, we can find what the population is for a given $t$.

Consider this: if we know the population but want to find the $t$ when the city reaches that population, we're basically asking “what $t$ fulfills that $n\cdot2^t=something$?” For your scenario $n\cdot2^t=3n$, we can simply change that to $2^t=3$.

Now, to directly access $t$'s value here, we can use the $\log$ function (as you probably know).

So, $2^t=3$ can be written as $t=\log_2(3)$. Since $t$ represents 50 years, the number of years it'll take to triple the population is simply $50t$. 🙂

Answer 2

The statement "under the assumption that the rate of increase is proportional to the number of inhabitants" means that the function number of inhabitants $P(t)$, where $t$ is the time (time unit = 1 year), solves the linear differential equation $$\frac{dP}{dt}(t)=c P(t)$$ where $c$ is the constant of proportionality. The general solution of such equation is $P(t)=P(0)\exp(ct)$ where $P(0)$ is the initial population. If $P(0)>0$ and $c>0$ then the population $P(t)$ grows exponentially.

Now "the population becoming double in 50 years" implies that $$P(t+50)=2P(t).$$ From this equation we can find the constant $c$: assuming $P(0)\not=0$, we get $$P(0)\exp(c(t+50))=2P(0)\exp(ct)\implies \exp(50c)=2\implies c=\frac{\ln(2)}{50}.$$ Finally we find $T$="number of years to triple" by solving $$P(t+T)=3P(t)$$ which yields $$P(0)\exp(c(t+T))=3P(0)\exp(ct)\implies \exp(cT)=3\\\implies T=\frac{\ln(3)}{c}=50\ln_2(3)\approx 79.25 years.$$

More generally, for $n\geq 1$, the equation $P(t+T_n)=nP(t)$ implies $T_n=50\ln_2(n)$, which means that at ANY time $t\geq 0$, given a population of $P(t)$, after $T_n$ years the population will be $n$ times $P(t)$.

Answer 3

This hints at the fact that the population was increasing at the rate of 2 percent per year.

This assumes a crude approximation at each timestep, and that percentages are additive, not multiplicative. (In fact, your logic will generally always result in this answer, regardless of how you divide up the time intervals.)

Suppose, in a strange world, that I am given two discounts on a $\$100$ bill: a $50\%$ discount, and a $25\%$ discount. Assume these are applied in order to the bill, and affect the total price of the bill post discounts.

Your logic seems to suggest a $75\%$ discount in the end, and a final bill of $\$25$. However, you would have $\$100 \cdot 0.5 \cdot 0.75 = \$37.50$ instead.

If you have a population of $100$ people that grows at $2\%$ per year, then you have $$ 100 \cdot 1.02 = 102 $$ people after one year; $$ 102 \cdot 1.02 = 100 \cdot 1.02^2 $$ after two years; $$ 100 \cdot 1.02^3 $$ after three, and so on. However, if you do this for $100$ years, you get $$ 100 \cdot 1.02^{100} \approx 724 $$ people. On the other hand, a population that grows by $2 \times 100\%=200\%$ after $100$ years only has $$ 100 \cdot 2 = 200 $$ people.

This is how exponential growth essentially works: growth based on a multiple of the previous population. Loosely, $$ P(t+\Delta t) \approx P(t) + \Delta t P'(t) $$ where $P$ is the population, and $t$ a time variable. (Assume $\Delta t$ is small.) Phrased differently, $$ \frac{dP}{dt} = k P $$ for some constant $k$: the rate at which the number grows is proportional to the value itself. Note in particular we have to use continuously varying quantities in this process.

In your model, you start with a population $P$, and poof $P/50$ people into existence the very moment $y$ years is hit, when in reality those people are having children the entire time -- and in theory, those children are themselves also having children, and their children are too. (Think of the Fibonacci bunnies.) You need to account for this continual growth more appropriately to get an accurate answer.

Another analogy you may find useful is the difference between interest compounded $n$ times per year, versus continuously-compounded interest.

Solve the differential equation given for the appropriate $k$ to figure out a closed form for $P(t)$, and then you can handle matters accordingly.

Answer 4

Simply put, you're assuming a linear rate of growth, while the problem asks you to assume an exponential rate of growth. Say for example the city starts with 10,000 people. Since it grows to 20,000 people in 50 years, the average rate of growth per year over that timespan (the slope of the secant line between the two data points) is 200 people per year. But it is a mistake to assume that the city grows at a constant and unchanging rate of growth of 200 people per year no matter how much time passes. Instead, the question asks you to model the population exponentially so that the rate of growth at any point in time is proportional to the population at that time. The model you have proposed is $P(t) = P + 0.02Pt$, which is an equation describing the population of a city that has a constant influx of the same amount of people every year equal to 2% of the original population. Instead, the correct model should be $P(t) = P \times 2^{(t/50)}$, which is an equation describing the population of a city that doubles in population every 50 years. Plugging in the appropriate values into this model yields the correct answer.

Answer 5

This means, that the population increases by 100 percent in 50 years. This hints at the fact that the population was increasing at the rate of 2 percent per year.

Wrong. If the population increased by 2 percent each year, then it would be $x$ at the start, then $1.02x$ after year one, $1.02^2$ after year $2$ and so on, and it would be $1.02^{50}x$ after $50$ years. Since $1.02^{50} \approx 2.69$, this population would more than double in $50$ years, which contradicts your assumptions.

Answer 6

A simple way is to envisage a compounding annual growth rate $R$, say, for the population,

then $R^{50} =2\tag 1$

the time $T$ we need is $R^T = 3\tag2$

and dividing $(2)$ by $(1)$ and taking logs of both and simplifying,

$T = 50\cdot\large\frac{log 3}{log 2}\,, \approx 79.25$ years