Question

Some time in the future a human colony is started on Mars. The colony begins with 30000 people and grows exponentially to 90000 in 200 years.

a. Give a formula for the size of the human population on Mars as a function of t= time (in years) since the founding of the original colony.
b. Assuming the population continues to grow exponentially, how long will it take to reach a size of 400000?
c. What is the rate of change of the size of the population 200 years after the founding of the original colony?

Answer 1

Answer:

a) $P(t) = 30000e^{0.0055t}$

b) It will take 470.5 years to reach a size of 400000

c) 495.69 people a year.

Step-by-step explanation:

The exponential population equation is as follows:

$P(t) = P(0)e^{rt}$

In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.

a. Give a formula for the size of the human population on Mars as a function of t= time (in years) since the founding of the original colony.

The colony begins with 30000 people, which means that $P(0) = 30000$

Grows exponentially to 90000 in 200 years. This means that $P(200) = 90000$, which helps us find r.

$P(t) = P(0)e^{rt}$

$90000 = 30000e^{200r}$

$e^{200r} = \frac{90000}{30000}$

$e^{200r} = 3$

Applying ln to both sides

$\ln{e^{200r}} = \ln{3}$

$200r = \ln{3}$

$r = \frac{\ln{3}}{200}$

$r = 0.0055$

So

$P(t) = 30000e^{0.0055t}$

b. Assuming the population continues to grow exponentially, how long will it take to reach a size of 400000?

This is t when $P(t) = 400000$. So

$P(t) = 30000e^{0.0055t}$

$400000 = 30000e^{0.0055t}$

$e^{0.0055t}= \frac{400000}{30000}$

$e^{0.0055t} = 13.3$

Applying ln to both sides

$\ln{e^{0.0055t}} = \ln{13.3}$

$0.0055t = \ln{13.3}$

$t = \frac{\ln{13.3}}{0.0055}$

$t = 470.5$

It will take 470.5 years to reach a size of 400000

c. What is the rate of change of the size of the population 200 years after the founding of the original colony?

This is $P'(200)$. So

$P(t) = 30000e^{0.0055t}$

$P'(t) = 30000*0.0055e^{0.0055t}$

$P'(t) = 165e^{0.0055t}$

$P'(200) = 165e^{0.0055*200} = 495.69$

Answer 2

A) The formula for the size of the human population on Mars as a function of t = time (in years) since the founding of the original colony is;

P(t) = 30000e^(0.0055t)

B) The time it will take for the population to reach a size of 400000 is;

t = 471 years

C) The rate of change of the size of the population 200 years after the founding of the original colony is;

P'(200) = 495.67

To solve this question, we will make use of the exponential equation which is;

P(t) = P₀e^(rt)

Where;

P(t) is the population after t years

P₀ is the initial population

r is the growth rate

t is the number of years

A) The colony begins with 30000 people. Thus;

P₀ = 30000

Grows to 90000 in 200 years. Thus;

P(200) = 90000

t = 200 years

Thus, equation is;

30000e^(200r) = 90000

e^(200r) = 90000/30000

e^(200r) = 3

200r = In 3

r = 1.09861/200

r = 0.0055

Thus, the formula will be;

P(t) = 30000e^(0.0055t)

B) We are told the population continues to grow exponentially and reaches a size of 400000.

Thus;

400000 = 30000e^(0.0055t)

e^(0.0055t) = 400000/30000

e^(0.0055t) = 13.3333

0.0055t = In 13.3333

0.0055t = 2.590267

t = 2.590267/0.0055

t = 470.96 years

Approximately; t = 471 years

C) We want to find the rate of change of population. Thus, we will find the derivative of P(t) to get;

P'(t) = (30000 × 0.0055)e^(0.0055t)

Thus;

P'(200) = (30000 × 0.0055)e^(0.0055 × 200)

P'(200) = 495.67

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