Question

plz help asap...The population of city A was 12,831 in 1900 and grew 5% per year for the next 10 years. For the same period, the function P(t) = 11,572(1.07)t models the population of city B, where P is the population and t is the time, in years since 1900

Answer 1

Explanation:

In order to answer these questions, let's first determine what the models for each city are. The form of these is the exponential function

`y=a(b)^x` where b will be replaced with (1 + r) which is the growth rate. a represents the initial population, b again is the growth rate and x is time in years.

For city A, the model is:

`y=12831(1.05)^x` where the initial population is 12,831 and grew at a rate of .05 per year.

For city B, the model is:

`y=11572(1.07)^x` where the initial population is 11,572 and grew at a rate of .07 per year.

The first question asks what was the population of each city in 1900?

We already know the population in city A was 12831 because it tells us it is. For city B, the wording says "...since 1900". That means that if we take this model for city B, we will plug in a 0 for x because no time has gone by. The population can be determined by raising 1.07 to the 0 power to get 1, and multiply that by the 11572 and get that the population in 1900 was 11572. You can also tell that from the model, because "a" is the initial population and we have a 11572 in that place. And obviously, A has the greater population in 1900.

The next question asks at what rate each is increasing by. A is increasing by 5%, which was given, and B was increasing by 7% which can be determined from the model. (1.07) was (1 + .07). Therefore, B is growing faster.

The last question asks what was the population of each in 1910 and which was greater.

If our models are based off the population in 1900, when x = 0, then in 1910, 10 years have gone by and x = 10 will be filled in into each x. For city A:

`y=12831(1.05)^{10]` and we have a population of 20900

For city B:

`y=11572(1.07)^(10)` and we have a population of 22763. B had the greater population in 1910. That makes sense according to the question that asked which city was growing faster. The growth rate of B was greater than that of A.