For number 12 you need to make an expression using the information given. a) 25g + 18r + 15b could be the expression.
b) The variables would be 25g for the 25 gold seats, 18r for the 18 red seats and 15b for the 15 blue seats.
c) Plug in the number given according to the variables.
25(100) + 18(250) + 15(200)
2500 + 4500 + 3000
7000 + 3000
$10,000
Answer:
The population in 40 years will be 1220.
Step-by-step explanation:
The population of a town grows at a rate proportional to the population present at time t.
This means that:
![P(t) = P(0)e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7Brt%7D)
In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.
The initial population of 500 increases by 25% in 10 years.
This means that ![P(0) = 500, P(10) = 1.25*500 = 625](https://tex.z-dn.net/?f=P%280%29%20%3D%20500%2C%20P%2810%29%20%3D%201.25%2A500%20%3D%20625)
We apply this to the equation and find t.
![P(t) = P(0)e^{rt}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20P%280%29e%5E%7Brt%7D)
![625 = 500e^{10r}](https://tex.z-dn.net/?f=625%20%3D%20500e%5E%7B10r%7D)
![e^{10r} = \frac{625}{500}](https://tex.z-dn.net/?f=e%5E%7B10r%7D%20%3D%20%5Cfrac%7B625%7D%7B500%7D)
![e^{10r} = 1.25](https://tex.z-dn.net/?f=e%5E%7B10r%7D%20%3D%201.25)
Applying ln to both sides
![\ln{e^{10r}} = \ln{1.25}](https://tex.z-dn.net/?f=%5Cln%7Be%5E%7B10r%7D%7D%20%3D%20%5Cln%7B1.25%7D)
![10r = \ln{1.25}](https://tex.z-dn.net/?f=10r%20%3D%20%5Cln%7B1.25%7D)
![r = \frac{\ln{1.25}}{10}](https://tex.z-dn.net/?f=r%20%3D%20%5Cfrac%7B%5Cln%7B1.25%7D%7D%7B10%7D)
![r = 0.0223](https://tex.z-dn.net/?f=r%20%3D%200.0223)
So
![P(t) = 500e^{0.0223t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20500e%5E%7B0.0223t%7D)
What will be the population in 40 years
This is P(40).
![P(t) = 500e^{0.0223t}](https://tex.z-dn.net/?f=P%28t%29%20%3D%20500e%5E%7B0.0223t%7D)
![P(40) = 500e^{0.0223*40} = 1220](https://tex.z-dn.net/?f=P%2840%29%20%3D%20500e%5E%7B0.0223%2A40%7D%20%3D%201220)
The population in 40 years will be 1220.
Answer:
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