The gross domestic product (GDP) of a country A is growing at a constant rate. In 2021, the GDP was 125 billion dollars, and in 2023 it was 155 billion dollars. At what percentage rate will the GDP grow in 2026?
This question came up in a calculus I assignment, not sure if I've come across a formula that explicitly calculates the GDP rate. I've thought of using the concept of exponential but I'm not really sure how to go about it?
I'm thinking use GDP from year 2021 as a "base" and then multiply e to get GDP in year 2023, but that's as far as i can get
To solve this problem, you can indeed use the concept of exponential growth. The formula for exponential growth is:
$$ P = P_0 \times e^{r t} $$
Where:
From the problem, we know that $P_0 = \$125$ billion dollars in 2021, and $P = \$155$ billion dollars in 2023. The time $t$ between 2021 and 2023 is 2 years. We can rearrange the formula to solve for the rate $r$:
$$ r = \frac{1}{t} \ln\left(\frac{P}{P_0}\right) $$
Let's calculate $r$ using this formula. After finding $r$, we can then use it to calculate the GDP in 2026, which is 5 years after 2021.
The growth rate $r$ is given by $\frac{\ln(31/25)}{2}$. Now, let's calculate the numerical value of this rate.
The annual growth rate $r$ is approximately $0.1076$ or $10.76\%$.
Now, to find the GDP in 2026, we use the same exponential growth formula. The time from 2021 to 2026 is 5 years. So, we calculate $P$ for $t = 5$ years:
$$ P = P_0 \times e^{r t} $$
Let's calculate the GDP for 2026 using this formula.
The GDP of country A in 2026, growing at an annual rate of approximately $10.76\%$, is expected to be about $214.02$ billion dollars.