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SVETLANKA909090 [29]
4 years ago
12

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 t

imes 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.
Mathematics
1 answer:
Whitepunk [10]4 years ago
7 0
For the answer to the question above,
f(x) = k*b^x 

<span>k is the value of your thing (in this case number of people) after 0 iterations. b would be the rate that the thing grows, in this case by 17%. x is the number of iterations (towns) that have been expanded to, hence how many times 1.17 is compounded. So our equation would look like this: </span>

<span>f(x) = 8(1.17)^x </span>

<span>simply plug in the number of towns he has. So in this case 16. </span>

<span>f(16) = 8(1.17)^16 </span>

<span>do the math (simplify) </span>

<span>f(16) = 98.64... </span>

<span>you can't have 0.64 of a person so we round down because that guy hasn't been "created" yet. </span>

<span>after 16 villages 98 people</span>
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Answer:

1. <em>Axis of symmetry</em>: x = 3

2. <em>Vertex:</em> (3,5)

3. <em>Solution of the equation</em>:

<u />

  • x-intercepts:

               (3-\sqrt{5},0) \\\\(3+\sqrt{5},0)

  • y-intercept: (0, -4)

Explanation:

<u>1. Equation:</u>

    y=-x^2+6x-4

<u>2. </u><em><u>Axis of symmetry:</u></em>

That is the equation of a parabola, whose standard form is:

                 y=ax^2+bx+c

Where:    

                 a=-1;b=6;c=-4

The axis of symmetry is the vertical line with equation:

               x=-b/2a

Substitute  a=-1,\text{ and }b=6

          x=-6/[(2)(-1)]=-6/(-2)=3

Thus, the axis of symmetry is:

            x=3

<em><u>3. Vertex</u></em>

<em><u /></em>

The x-coordinate of the vertex is equal to the axys of symmetry, i.e x = 3.

To find the y-xoordinate, substitute this value of x into the equation for y:

               y=-x^2+6x-4\\\\y=-(3)^2+6(3)-4=-9+18-4=5

Therefore, the vertex is (3, 5)

<u>4. Find the x-intercepts</u>

The x-intercepts are the roots of the equation, which are the points wher y = 0.

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Use the quadratic equation:

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<u>5. Find the y-intercept</u>

<u />

The y-intercet is the value of y when x=0:

         y=-x^2+6x-4\\\\y=-(0)^2+6(0)-4=-4

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