Question

In 1987, the population of country A was estimated at 87 million people, with an annual growth rate of 3.5%. The 1987 population of country B was estimated at 243 million with an annual growth rate of 0.6%. Assume that both populations are growing exponentially. (Round your answers to the nearest whole number.) (a) In what year will country A double its 1987 population?(b) In what year will the the two countries have the same population?

Answer 1

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

Answer 2

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

$P(A) = 87e^{0.035t}$

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

$P(B) = 243e^{0.006t}$ where t = time in years and P in millions

a) P(A) = 2*87 when

$2=e^{0.035t}$

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

$P(A) = 87e^{0.035t}=P(B) = 243e^{0.006t}\\e^{0.035t-0.006t}=\frac{243}{87} =2.793\\0.029t = 1.027\\t= 35.42$

Approximately after 35.5 years the populations of both countries would be equal.