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?

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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a)

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:

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

Grows exponentially to 90000 in 200 years. This means that , which helps us find r.

Applying ln to both sides

So

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

This is t when . So

Applying ln to both sides

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 . So