Sure! Let's break down the given problem step by step. To find the population at a certain time given exponential growth, we'll use the exponential growth formula. The formula for exponential growth is:
[tex]\[ N(t) = N(0) \cdot e^{(r \cdot t)} \][/tex]
Where:
- [tex]\( N(t) \)[/tex] is the population at time [tex]\( t \)[/tex].
- [tex]\( N(0) \)[/tex] is the initial population size.
- [tex]\( e \)[/tex] is the base of natural logarithms (approximately 2.71828).
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the time period.
Given values:
- Initial population [tex]\( N(0) \)[/tex] = 10
- Growth rate [tex]\( r \)[/tex] = 1 individual per unit time
- Time period [tex]\( t \)[/tex] = 25 units
We will substitute these values into the exponential growth equation.
[tex]\[ N(t) = 10 \cdot e^{(1 \cdot 25)} \][/tex]
First, we calculate the exponent:
[tex]\[ 1 \cdot 25 = 25 \][/tex]
Next, we raise [tex]\( e \)[/tex] to the power of 25:
[tex]\[ e^{25} = 72004899.33738588 \][/tex]
(N.B. This value is based on a mathematical constant, and its computation can be typically done using a scientific calculator or computational tools.)
Now, multiply this result by the initial population [tex]\( N(0) \)[/tex]:
[tex]\[ N(25) = 10 \cdot 72004899.33738588 = 720048993.3738588 \][/tex]
So, the population after 25 units of time will be approximately:
[tex]\[ 720048993.3738588 \][/tex]
Therefore, the population will be approximately [tex]\( 720048993373.8588 \)[/tex] individuals at [tex]\( t = 25 \)[/tex] units of time.
