Question

Brian has been playing a game where he can create towns and help his empire expand. Each town he has allows him to create 1.17 times as many villagers. The game gave Brian eight villagers to start with. Help Brian expand his empire by solving for how many villagers he can create with 16 towns. Then explain to Brian how to create an equation to predict the number of villagers for any number of towns. Show your work and use complete sentences.

Answer 1

Let \(a_i =\) number of villagers when there are \(i\) towns.
Initially, there are \(a_0=8\) villagers.
$$
\large{
a_0=8\\
a_1=a_0\times 1.17\\
a_2=a_1\times1.17=a_0\times1.17^2\\
...\\\
a_i=a_0\times1.17^i
}
$$
We need then number of villagers when there are \(i=16\) towns.

Answer 2

Thank you so much. Would you mind helping me with another?

Answer 3

You're welcome.

Are you studying recursive sequences? The approach to these type problems can vary depending on what you are studying.

Answer 4

Victoria has some money from her birthday and the amount is modeled by the function h(x) = 200. She read about a bank that has savings accounts that accrue interest according to the function s(x) = (1.05)x − 1. Victoria is thinking about putting her money into the savings account to gain interest. Using complete sentences, explain to Victoria how she can combine her functions to create a new function, and explain what this new function means.

Yes, and I seriously don't get Algebra word problems.

Answer 5

Let \(x\) represent time, say month x=1,x=2, ... and so on.

So h(x)=200 means at any point in time (x), she has the same amount of money, $200. It never changes. It doesn't grow, it doesn't shrink because as x changes,going from 1, 2 , .. and so on, h(x) never changes, it really doesn't depend at all on month. We say that h(x) is independent of x.

If she put this $200 in a savings account, then her savings would grow by
$$
s(x)\times h(x)\\
=(1.05x -1)\times h(x)\\
=(1.05x-1)\times 200\\
=1.05\times 200x-200\\
=210\times x - 200
$$
So for example, on month x=1, the 1st month, her savings would grow
$$
210\times1-200=$10.
$$
And she would have $210 in her account instead of just $200.

Answer 6

I seriously appreciate your time helping me with this.

Answer 7

you're welcome