City A had a population of 10000 in the year 1990. City A’s population grows at a constant rate of 3% per year. City B has a population that is growing exponentially. In the year 2000, there were half as many people in B as in A. In the year 2010, the population of A was 20% more than the population of B. When will the populations be equal? Give your answer in years after 1990.

City A and city B will have equal population 25years after 1990

Step-by-step explanation:

Given

Let

years after 1990

population function of city A

population function of city B

City A

---- initial population (1990)

--- rate

City B

----- t = 10 in 2000

---- t = 20 in 2010

Required

When they will have the same population

Both functions follow exponential function.

So, we have:

Calculate the population of city A in 2000 (t = 10)

Calculate the population of city A in 2010 (t = 20)

From the question, we have:

 and  

Solve for B20

and can be used to determine the function of city B

For:

We have:

For:

We have:

Divide by

Apply law of indices

--- (1)

Take 10th root of both sides

Subtract 1 from both sides

To calculate , we have:

Recall that:

So:

Hence:

The question requires that we solve for t when:

Where:

and

becomes

Divide both sides by 10000

Divide both sides by

Take natural logarithm of both sides

Rewrite as:

Solve for t

Approximate

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

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whats the