Respuesta :
Answer:
1). [tex]P_{t}=320\times 2^{(t-1)}[/tex]
2). 40960
Step-by-step explanation:
Initial population of the mouse has been given = 320
The mouse population is doubling every year.
So, population after one year = 320 × 2
Population after second year = 320 × 2²
Population after third year = 320 × 2³
So the sequence formed is 320, 320×2, 320×4...........
We can see population after every year is increasing exponentially with a common factor = [tex]\frac{320\times 2^{2} }{320\times 2}[/tex] = 2
Part 1.
Now we know the formula which models the population increase will be
[tex]P_{t}=P_{0}r^{(t-1)}[/tex]
Where [tex]P_{t}[/tex] = Population after t years
and [tex]P_{0}[/tex] = Initial population
[tex]P_{t}=320\times 2^{(t-1)}[/tex]
Part 2.
We have to calculate the value of [tex]P_{8}[/tex].
[tex]P_{8}=320\times 2^{(8-1)}[/tex]
= [tex]320\times 2^{7}[/tex]
= 320 × 128
= 40960
Exponential functions are often used to model population.
- The exponential function is [tex]\mathbf{P(t) = 320 \times 2^{t-1}}[/tex]
- The mouse population after 8 years is 40960
The given parameters are:
[tex]\mathbf{P_1 = 320}[/tex] --- initial population
[tex]\mathbf{r = 2}[/tex] -- rate
(a) The exponential function
This is calculated as:
[tex]\mathbf{P(t) = P_1 \times r^{t-1}}[/tex]
Substitute known values
[tex]\mathbf{P(t) = 320 \times 2^{t-1}}[/tex]
(b) The population after 8 years
This means that t = 8.
So, we have:
[tex]\mathbf{P(8) = 320 \times 2^{8-1}}[/tex]
[tex]\mathbf{P(8) = 320 \times 2^7}[/tex]
[tex]\mathbf{P(8) = 320 \times 128}[/tex]
[tex]\mathbf{P(8) = 40960}[/tex]
Hence, the mouse population after 8 years is 40960
Read more about exponential functions at:
https://brainly.com/question/11487261
