How do we solve these two problems?

3. A tank contains 100 gal of water and 50 oz of salt. Water containing a salt concentration of i (1 + % sin t) oz / gal ows into the tank at a rate of 2 gal/ min, and the mixture in the tank ows out at the same rate. a. Find the amount of salt in the tank at any time. b. Plot the solution for a time period long enough so that you see the ultimate behavior of the graph. c. The longtime behavior of the solution is an oscillation about a certain constant level. What is this level? What is the amplitude of the oscillation? 63,150 _ . 3; a.Q(t)=We t/50+25% cos t+% smt c. level = 25; amplitude = 25\\/ 2501/5002 E 0.24995 4. Suppose that a tank containing a certain liquid has an outlet near the bottom. Let h(t) be the height of the liquid surface above the outlet at time t. Ton'icelli'sz principle states that the outow velocity v at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height h. a. Show that v = 1 /2gh, where g is the acceleration due to gravity. b. By equating the rate of outow to the rate of change of liquid in the tank, show that h(t) satises the equation (34) dh A (h) E =aa' gh, where A(h) is the area of the cross section of the tank at height h and a is the area of the outlet. The constant a is a contraction coefcient that accounts for the observed fact that the cross section of the (smooth) outow stream is smaller than a. The value of a for water is about 0.6. 0. Consider a water tank in the form of a right circular cylinder that is 3 m high above the outlet. The radius of the tank is 1 m, and the radius of the circular outlet is 0.1 m. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet. 4; c. 130.41 s