Under ideal conditions, the population of a certain species doubles every nine years. If the population starts with 100 individuals, which of the following expressions would give t... Under ideal conditions, the population of a certain species doubles every nine years. If the population starts with 100 individuals, which of the following expressions would give the population of the species t years after the start, assuming that the population is living under ideal conditions?
Understand the Problem
The question is asking us to find an expression for the population of a species that doubles every nine years, starting from 100 individuals. We need to determine which of the given expressions correctly represents this growth after 't' years.
Answer
The population after $t$ years is given by the expression $100 \times 2^{t/9}$.
Answer for screen readers
The correct expression for the population after $t$ years is:
$$ P(t) = 100 \times 2^{t/9} $$
Steps to Solve
-
Identify the Growth Pattern
The population doubles every 9 years. This can be expressed mathematically using the formula for exponential growth. -
Write the Exponential Growth Formula
The general formula for population growth is:
$$ P(t) = P_0 \times (2^{t/T}) $$
Where:
- $P(t)$ = population after time $t$
- $P_0$ = initial population
- $T$ = time taken to double the population (in this case, 9 years)
-
Substitute the Known Values
Substituting the initial population $P_0 = 100$ and the doubling time $T = 9$:
$$ P(t) = 100 \times (2^{t/9}) $$ -
Simplifying the Exponential Expression
The expression $2^{t/9}$ can be rewritten as:
$$ P(t) = 100 \times 2^{\frac{t}{9}} $$ -
Compare with Given Options
Now, check this expression against the given options. We find that:
$$ P(t) = 100 \times 2^{\frac{t}{9}} $$
matches the option $100 \times 2^{\frac{t}{9}}$.
The correct expression for the population after $t$ years is:
$$ P(t) = 100 \times 2^{t/9} $$
More Information
This type of problem demonstrates the concept of exponential growth, which is commonly observed in populations under ideal conditions. When a population doubles at a constant rate, it shows a consistent multiplicative growth pattern over equal time intervals.
Tips
- Forgetting to use the initial population of 100 individuals in the formula.
- Miscalculating the exponent or the base related to the doubling time.
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