Volume 48
6x 8y
Multiple 6 times 8
Answer:
Runner 1 has the fastest pace
Step-by-step explanation:
Answer:
your welcome it has to be c
Answer:
The country's population for the year 2030 is 368.8 million.
Step-by-step explanation:
The differential equation is:
![\frac{dP}{dt}=kP(600 - P)\\\frac{dP}{P(600 - P)} =kdt](https://tex.z-dn.net/?f=%5Cfrac%7BdP%7D%7Bdt%7D%3DkP%28600%20-%20P%29%5C%5C%5Cfrac%7BdP%7D%7BP%28600%20-%20P%29%7D%20%3Dkdt)
Integrate the differential equation to determine the equation of P in terms of <em>t</em> as follows:
![\int\limits {\frac{1}{P(600-P)} } \, dP =k\int\limits {1} \, dt \\(\frac{1}{600} )[(\int\limits {\frac{1}{P} } \, dP) - (\int\limits {\frac{}{600-P} } \, dP)]=k\int\limits {1} \, dt\\\ln P-\ln (600-P)=600kt+C\\\ln (\frac{P}{600-P} )=600kt+C\\\frac{P}{600-P} = Ce^{600kt}](https://tex.z-dn.net/?f=%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7BP%28600-P%29%7D%20%7D%20%5C%2C%20dP%20%3Dk%5Cint%5Climits%20%7B1%7D%20%5C%2C%20dt%20%5C%5C%28%5Cfrac%7B1%7D%7B600%7D%20%29%5B%28%5Cint%5Climits%20%7B%5Cfrac%7B1%7D%7BP%7D%20%7D%20%5C%2C%20dP%29%20-%20%28%5Cint%5Climits%20%7B%5Cfrac%7B%7D%7B600-P%7D%20%7D%20%5C%2C%20dP%29%5D%3Dk%5Cint%5Climits%20%7B1%7D%20%5C%2C%20dt%5C%5C%5Cln%20P-%5Cln%20%28600-P%29%3D600kt%2BC%5C%5C%5Cln%20%28%5Cfrac%7BP%7D%7B600-P%7D%20%29%3D600kt%2BC%5C%5C%5Cfrac%7BP%7D%7B600-P%7D%20%3D%20Ce%5E%7B600kt%7D)
At <em>t</em> = 0 the value of <em>P</em> is 300 million.
Determine the value of constant C as follows:
![\frac{P}{600-P} = Ce^{600kt}\\\frac{300}{600-300}=Ce^{600\times0\times k}\\\frac{1}{300} =C\times1\\C=\frac{1}{300}](https://tex.z-dn.net/?f=%5Cfrac%7BP%7D%7B600-P%7D%20%3D%20Ce%5E%7B600kt%7D%5C%5C%5Cfrac%7B300%7D%7B600-300%7D%3DCe%5E%7B600%5Ctimes0%5Ctimes%20k%7D%5C%5C%5Cfrac%7B1%7D%7B300%7D%20%3DC%5Ctimes1%5C%5CC%3D%5Cfrac%7B1%7D%7B300%7D)
It is provided that the population growth rate is 1 million per year.
Then for the year 1961, the population is: P (1) = 301
Then
.
Determine <em>k</em> as follows:
![\frac{dP}{dt}=kP(600 - P)\\1=k\times300(600-300)\\k=\frac{1}{90000}](https://tex.z-dn.net/?f=%5Cfrac%7BdP%7D%7Bdt%7D%3DkP%28600%20-%20P%29%5C%5C1%3Dk%5Ctimes300%28600-300%29%5C%5Ck%3D%5Cfrac%7B1%7D%7B90000%7D)
For the year 2030, P (2030) = P (70).
Determine the value of P (70) as follows:
![\frac{P(70)}{600-P(70)} = \frac{1}{300} e^{\frac{600\times 70}{90000}}\\\frac{P(70)}{600-P(70)} =1.595\\P(70)=957-1.595P(70)\\2.595P(70)=957\\P(70)=368.786](https://tex.z-dn.net/?f=%5Cfrac%7BP%2870%29%7D%7B600-P%2870%29%7D%20%3D%20%5Cfrac%7B1%7D%7B300%7D%20e%5E%7B%5Cfrac%7B600%5Ctimes%2070%7D%7B90000%7D%7D%5C%5C%5Cfrac%7BP%2870%29%7D%7B600-P%2870%29%7D%20%3D1.595%5C%5CP%2870%29%3D957-1.595P%2870%29%5C%5C2.595P%2870%29%3D957%5C%5CP%2870%29%3D368.786)
Thus, the country's population for the year 2030 is 368.8 million.