Answer:
The population would be 34007.
Step-by-step explanation:
The given equation that shows the population growth rate of a certain city,
![r(t)=1800(1+\frac{6t}{3+t^2}); 0\leq t\leq 5](https://tex.z-dn.net/?f=r%28t%29%3D1800%281%2B%5Cfrac%7B6t%7D%7B3%2Bt%5E2%7D%29%3B%200%5Cleq%20t%5Cleq%205)
Let P represents the population,
Since, change rate in population = r(t)
![\frac{dP(t)}{dt}=r(t)](https://tex.z-dn.net/?f=%5Cfrac%7BdP%28t%29%7D%7Bdt%7D%3Dr%28t%29)
![\frac{dP(t)}{dt}=1800(1+\frac{6t}{3+t^2})](https://tex.z-dn.net/?f=%5Cfrac%7BdP%28t%29%7D%7Bdt%7D%3D1800%281%2B%5Cfrac%7B6t%7D%7B3%2Bt%5E2%7D%29)
![dP(t)=1800(1+\frac{6t}{3+t^2})dt](https://tex.z-dn.net/?f=dP%28t%29%3D1800%281%2B%5Cfrac%7B6t%7D%7B3%2Bt%5E2%7D%29dt)
Integrating both sides,
![\int dP(t)=\int 1800(1+\frac{6t}{3+t^2})dt](https://tex.z-dn.net/?f=%5Cint%20dP%28t%29%3D%5Cint%201800%281%2B%5Cfrac%7B6t%7D%7B3%2Bt%5E2%7D%29dt)
![P(t)=\int 1800 dt+\int \frac{6t}{3+t^2}dt](https://tex.z-dn.net/?f=P%28t%29%3D%5Cint%201800%20dt%2B%5Cint%20%5Cfrac%7B6t%7D%7B3%2Bt%5E2%7Ddt)
![P(t)=1800t + \int \frac{6t}{3+t^2}dt](https://tex.z-dn.net/?f=P%28t%29%3D1800t%20%2B%20%5Cint%20%5Cfrac%7B6t%7D%7B3%2Bt%5E2%7Ddt)
Put ![3+t^2 = u\implies 2t dt = du](https://tex.z-dn.net/?f=3%2Bt%5E2%20%3D%20u%5Cimplies%202t%20dt%20%3D%20du)
![P(t)=1800t + \int \frac{3}{u}du](https://tex.z-dn.net/?f=P%28t%29%3D1800t%20%2B%20%5Cint%20%5Cfrac%7B3%7D%7Bu%7Ddu)
![P(t)=1800t + 3\ln u + C](https://tex.z-dn.net/?f=P%28t%29%3D1800t%20%2B%203%5Cln%20u%20%2B%20C)
![P(t)=1800t + 3\ln (3+t^2) + C](https://tex.z-dn.net/?f=P%28t%29%3D1800t%20%2B%203%5Cln%20%283%2Bt%5E2%29%20%2B%20C)
We have P(0) = 25000
![25000 = 1800(0) + 3\ln (3) + C](https://tex.z-dn.net/?f=25000%20%3D%201800%280%29%20%2B%203%5Cln%20%283%29%20%2B%20C)
![25000 = 3\ln (3) + C](https://tex.z-dn.net/?f=25000%20%3D%203%5Cln%20%283%29%20%2B%20C)
![\implies C = 25000 - 3\ln (3) =24996.70](https://tex.z-dn.net/?f=%5Cimplies%20C%20%3D%2025000%20-%203%5Cln%20%283%29%20%3D24996.70)
Hence, the function that shows the population after t years,
![P(t)=1800t + 3\ln (3+t^2) + 24996.70](https://tex.z-dn.net/?f=P%28t%29%3D1800t%20%2B%203%5Cln%20%283%2Bt%5E2%29%20%2B%2024996.70)
At t = 5,
Population after 5 years would be,
![P(5) = 1800(5) + 3\ln (3+5^2) + 24996.70 = 34006.70\approx 34007](https://tex.z-dn.net/?f=P%285%29%20%3D%201800%285%29%20%2B%203%5Cln%20%283%2B5%5E2%29%20%2B%2024996.70%20%3D%2034006.70%5Capprox%2034007)
Answer:
The value of the digit 6 in 23,640 is ? Times ? Than the value of the 6 in 167892
Answer:
x = 13/25
Step-by-step explanation:
- 4 (9 - 7x) - (1 - x) = 4 (x - 6)
- 36 + 28x - 1 + x = 4x - 24
28x + x - 4x = - 24 + 36 + 1
25x = 13
x = 13/25