SOLUTION: Please help, The volume of water in a tank at a particular time (measured in seconds) is given by V (t) = 5t(2 − t) m3. Find the rate of change of the volume of water in t

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Please help, The volume of water in a tank at a particular time (measured in seconds) is given by V (t) = 5t(2 − t) m3. Find the rate of change of the volume of water in t      Log On


   

Question 968480: Please help,
The volume of water in a tank at a particular time (measured in seconds) is given by V (t) = 5t(2 − t) m3. Find the rate of change of the volume of water in the tank from first principles (using the definition of the rate of change).
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
During the time, dt,
dV%2Fdt=%28V%28t%2Bdt%29-V%28t%29%29%2Fdt
V%28t%2Bdt%29=5%28t%2Bdt%29%282-%28t%2Bdt%29%29
V%28t%2Bdt%29=%285t%2B5dt%29%282-t-dt%29
V%28t%2Bdt%29=%2810t-5t%5E2-5tdt%29%2B%2810dt-5tdt-5%28dt%29%5E2%29
V%28t%2Bdt%29=10t-5t%5E2-10tdt%2B10dt-5%28dt%29%5E2
So then,
V%28t%2Bdt%29-V%28t%29=10t-5t%5E2-10tdt%2B10dt-5%28dt%29%5E2-10t-5t%5E2
V%28t%2Bdt%29-V%28t%29=-10tdt%2B10dt-5%28dt%29%5E2
and,
%28V%28t%2Bdt%29-V%28t%29%29%2Fdt=-10t%2B10-5dt
But dt becomes infinitesimally small so,
%28V%28t%2Bdt%29-V%28t%29%29%2Fdt=-10t%2B10
dV%2Fdt=-10t%2B10