Two tanks are connected as shown in Figure 10.6. Tank 1 initially contains 100 gallons of water in which 40 pounds of salt are dissolved. Tank 2 initially contains 150 gallons of pure water. Beginning at t = 0, a brine solution containing 1/5 pound of salt per gallon is pumped into tank 1 at the rate of 5 gallons per minute. At this time, a solution which also contains 1/5 pound of salt per gallon is pumped into tank 2 at the rate of 10 gallons per minute. The brine solutions are interchanged between the tanks and also flow out of both tanks at the rates shown. Determine the amount of salt in each tank for t 0. Also calculate the time at which the brine solution in tank 1 reaches its minimum salinity (concentration of salt) and determine how much salt is in tank 1 at that time.
Calculus notes of 9/26/16 3.11 Related Rates Suppose a spherical weather balloon is filled with gas at a rate of 150 ft /min. What is the rate of change of its radius when the radius is 20 ft We are given / = 150: Volume increases by 150 ft /min. 3 dt dr We want / whedtr = 20 We need an equation relating V to r. 3 For a sphere, V = 4/3ᴨr dv 2 dr Different Rating time / = 4/3dt* 3r / (implicitdtifferentiation) 2 Or V’ = 4/3ᴨ * 3r r’ Solving for / , dt = dt(4ᴨr ) * /2 dvdt = 1/(4ᴨ * 20 ) * 150 = (3/32ᴨ) ft/min = 0.02984 ft/min Procedure: 1. Identify known rates and desired rates. Label what you alread