Respuesta :
Answer:
Step-by-step explanation:
Given that in 1987, the population of country A was estimated at 87 million people, with an annual growth rate of 3.5%
Thus the equation for population for country A would be
[tex]P(A) = 87e^{0.035t}[/tex]
The 1987 population of country B was estimated at 243 million with an annual growth rate of 0.6%.
So equation for population for country B would be
[tex]P(B) = 243e^{0.006t}[/tex] where t = time in years and P in millions
a) P(A) = 2*87 when
[tex]2=e^{0.035t}[/tex]
Take log and solve
ln 2 = 0.005 t
t = 138.63
Thus after 138 years population will double for A
b) P(A) = P(B) when
[tex]P(A) = 87e^{0.035t}=P(B) = 243e^{0.006t}\\e^{0.035t-0.006t}=\frac{243}{87} =2.793\\0.029t = 1.027\\t= 35.42[/tex]
Approximately after 35.5 years the populations of both countries would be equal.
Answer:
Step-by-step explanation:
Since both populations are growing exponentially, we would apply the formula for determining exponential growth which is expressed as
A = P(1 + r)^t
Where
A represents the population after t years.
t represents the number of years.
P represents the initial population.
r represents rate of growth.
a) for the population to double,
A = 2(87 × 10^6) = 174 × 10^6
P = 87 × 10^6
r = 3.5%% = 3.5/100 = 0.035
Therefore
174 × 10^6 = 87 × 10^6(1 + 0.035)^t
174 × 10^6/87 × 10^6 = (1 + 0.035/1)^t
2 = (1.035)^t
Taking log of both sides to base 10
Log 2 = log1.035^t = tlog1.035
0.3010 = t × 0.015
t = 0.3010/0.015 = 20 years
The year would be
1987 + 20 = 2007
b) let t represent the year when the the two countries will have the same population. Therefore,
In t years, the population of country A would be
87 × 10^6(1 + 0.035)^t = 87 × 10^6(1.035)^t
In t years also, the population of country B would be
243 × 10^6(1 + 0.006)^t = 243 × 10^6(1.006)^t
For both populations to be the same, the number of years that it will take would be
87 × 10^6(1.035)^t = 243 × 10^6(1.006)^t
87(1.035)^t = 243(1.006)^t
By iterating,
t is approximately 36 years
