Question

The rate of increase of population of a city is proportional to population present. In 40 years it increased from 30,000 to 40,000. At time $t$ population is $a(b)^{t/40}$. Then $a$ and $b$ are \dots}

• A. 30,000, 2/3
• B. 30,000, 4/3
• C. 40,000, 2/3
• D. 40,000, 3/4

Hint:

In any exponential growth/decay formula $y(t) = a \cdot b^{kt}$, the constant '$a$' represents the initial starting amount (when $t=0$), and '$b$' represents the growth/decay multiplier factor over the specific time interval defined in the exponent.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Exponential growth is represented by $P = P_0 e^{kt}$. The form given is $P = a(b)^{t/40}$.

Step 2: Formula Application:
At $t = 0, P = 30,000$. $30,000 = a(b)^0 \implies a = 30,000$.

Step 3: Explanation:
At $t = 40, P = 40,000$. $40,000 = 30,000(b)^{40/40}$ $40,000 = 30,000(b)^1$ $b = 40,000/30,000 = 4/3$.

Step 4: Final Answer:
The values are $a = 30,000$ and $b = 4/3$.