Respuesta :
Answer:
Step-by-step explanation:
At the time t = 0, population of the town = 5000
Rate of population increase = 500 per year
Therefore, the equation that will represent the population will be
[tex]P_{t}=P_{0}+500t[/tex]
Where [tex]P_{t}[/tex] = Population after t years
[tex]P_{0}[/tex]= Initial population
t = Time in years
a). For double once the population will be 500×2 = 10000
By plugging in the values in the equation,
10000 = 5000 + 500t
500t = 10000 - 5000
500t = 5000
t = [tex]\frac{5000}{500}[/tex]
t = 10 years
For Double twice,
Population will be = 10000×2 = 20000
Now we plug in the values in the equation again
20000 = 5000 + 500t
500t = 20000 - 5000
500t = 15000
t = [tex]\frac{15000}{500}[/tex]
t = 30 years
For double thrice,
Population of the town = 20000×2 = 40000
Now we plug in the values in the equation,
40000 = 5000 + 500t
500t = 40000 - 5000
500t = 35000
t = [tex]\frac{35000}{500}[/tex]
t = 70 years
b). If the population growth is 5%.
Then the growth will be exponential represented by
[tex]T_{n}=T_{0}(1+\frac{r}{100})^{t}[/tex]
[tex]T_{n}[/tex] = Population after t years
[tex]T_{0}[/tex] = Initial population
t = time in years
For double once,
Population after t years = 10000
[tex]10000=5000(1+\frac{5}{100})^{t}[/tex]
[tex](1.05)^{t}=\frac{10000}{5000}[/tex]
[tex](1.05)^{t}=2[/tex]
Take log on both the sides
[tex]log(1.05)^{t}=log2[/tex]
tlog(1.05) = log2
t = [tex]\frac{log2}{log1.05}[/tex]
t = 14.20 years
For double twice,
Population after t years = 20000
[tex]20000=5000(1+\frac{5}{100})^{t}[/tex]
[tex](1.05)^{t}=\frac{20000}{5000}[/tex]
[tex](1.05)^{t}=4[/tex]
Take log on both the sides
[tex]log(1.05)^{t}=log4[/tex]
tlog(1.05) = log4
t = [tex]\frac{log4}{log1.05}[/tex]
t = 28.413 years
For double thrice
Population after t years = 40000
[tex]40000=5000(1+\frac{5}{100})^{t}[/tex]
[tex](1.05)^{t}=\frac{40000}{5000}[/tex]
[tex](1.05)^{t}=8[/tex]
Take log on both the sides
[tex]log(1.05)^{t}=log8[/tex]
tlog(1.05) = log8
t = [tex]\frac{log8}{log1.05}[/tex]
t = 42.620 years
