Question Statement

The population of the world was 5.2 billion in 1990. The exponential growth rate was $1.6\%$ per year at that time:

(a) Find the exponential growth function. (b) Find the population of the world in 2000. (c) In which year was the world population 8 billion?

---

Background and Explanation

Exponential growth is used to model populations that increase by a fixed percentage over regular time intervals. The general formula is $P(t)=P_0(1+r{)}^t$, where $P_0$ is the initial population, $r$ is the growth rate as a decimal, and $t$ is the time elapsed.

---

Solution

Part (a)

To find the exponential growth function, we first identify the initial population ($P_0$) and the growth rate ($r$).

• Initial population in 1990: $P_0=5.2$ billion.
• Growth rate: $1.6\%=\frac{1.6}{100}=0.016$.

The growth factor is calculated as $(1+r)=1+0.016=1.016$. Substituting these values into the general formula $P(t)=P_0(1+r{)}^t$, we get: $P(t)=5.2(1.016{)}^t$

Part (b)

We need to find the population in the year 2000. First, calculate the number of years ($t$) since 1990: $t=2000-1990=10\text{ years}$

Now, substitute $t=10$ into our growth function: $P(10)=5.2(1.016{)}^{10}$ $P(10)\approx 5.2(1.172)$ $P(10)\approx 6.094\text{ billion}$

Part (c)

We are given that the population $P(t)=8$ billion and we need to solve for $t$: $8=5.2(1.016{)}^t$

First, isolate the exponential term by dividing both sides by 5.2: $\frac{8}{5.2}=(1.016{)}^t$

To solve for the exponent $t$, we take the logarithm of both sides: $\log \left(\frac{8}{5.2}\right)=\log (1.016^t)$ $\log 8-\log 5.2=t\log 1.016$

Now, solve for $t$: $t=\frac{\log 8-\log 5.2}{\log 1.016}$ $t\approx \frac{0.9031-0.7160}{0.00689}$ $t\approx 27.14\text{ years}$

To find the specific year, add this value to the starting year (1990): $\text{Year}=1990+27.14=2017.14$

So, the world population reached 8 billion during the year 2017.

---

Key Formulas or Methods Used

• Exponential Growth Formula: $P(t)=P_0(1+r{)}^t$
• Growth Factor: $1+r$
• Logarithm Power Rule: $\log (a^b)=b\log a$
• Logarithm Quotient Rule: $\log (\frac{a}{b})=\log a-\log b$

---

Summary of Steps

1. Identify the initial value ($5.2$) and convert the percentage rate to a decimal ($0.016$).
2. Construct the function $P(t)=5.2(1.016{)}^t$.
3. Calculate $t$ for the target year and solve the equation to find the future population.
4. Set the population to 8 and use logarithms to solve for the unknown time $t$.
5. Add the resulting time to the base year (1990) to determine the calendar year.