Question

The population of a city (in millions) at time t (in years) is P(t)=2.6 e 0.005t , where t=0 is the year 2000. When will the population double from its size at t=0 ?

Answer 1

Answer:

  year 2139

Step-by-step explanation:

The population will double when the factor e^(.005t) is 2.

  e^(.005t) = 2

  .005t = ln(2)

  t = ln(2)/0.005 = 138.6

The population will be double its size at t=0 when t=138.6. That is the population will be about 5.2 million in the year 2139.

Answer 2

The population will double by the year 2139 from its value of 2.6 million in year 2000.

Population function :

$P(t) = 2.6 {e}^{0.005t}$

Population size at t = 0

$P(0) = 2.6 {e}^{0.005(0)} = 2.6(1) = 2.6$

Population at t = 2.6 million.

For the population to double ;

2.6 × 2 = 5.2 million :

$5.2 = 2.6 {e}^{0.005t}$

We solve for t

$\frac{5.2}{2.6} = {e}^{0.005t}$

$2 = {e}^{0.005t}$

Take the In of both sides

$ln(2) = 0.005t$

$t \: = ln(2) \div 0.005 = 138.629$

The population will double after 139 years

Therefore, the population will double by the 2139 (Year 2000 + 139 years) = year 2139.

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