Question

Easy question

water flows continuously froma large tank at a rate proportional to the amount of water remaining; that is, dy/dt=ky. ther was initially 10000 cubic feet of water in the tank. at t=4 hours, 8000 cubic feet of water remained, what is k in dy/dt=ky?

show all work

Answer 1

`(\mathrm dy)/(\mathrm dt)=ky\iff\frac{\mathrm dy}y=k\,\mathrm dt`

Integrating both sides, we get


`\ln|y|=kt+C`

`\implies y=e^(kt+C)=e^(kt)e^C=Ce^(kt)`

When
`t=0`, we have
`y=10000`, so that


`10000=Ce^(0k)\implies C=10000`

When
`t=4`, we have
`y=8000`, which means


`8000=10000e^(4k)\implies k=\frac14\ln(8000)/(10000)\approx-0.0558`