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# Derivatives and Differential Equations and Leaking Tank Word Problem

Water is pumped_into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of &#8730;(t+1) gallons per minute for 0 &#8804; r &#8804; 120 minutes. At time t = 0, the tank contains 30 gallons water.
(a) How many gallons of water leak out of the tank from time r = 0 to r = 3 minutes?
(b) How many gallons of water are in the tank at time t = 3 minutes?
(c) Write an expression for A(r), the total number of gallons of water in the tank at tune r.
(d) At what time t, for 0 &#8804; t &#8804; 120 is the amount of water in the tank a maximum? Justify your answer.

Consider the curve given by....
(a) Show that...
(b) Find all points on the curve whose x-coordinate is 1, and write an equation for the tangent at each of these points.
(c) Find the x-coordinate of each point on the curve where the tangent line is vertical.

Consider the differential equation....
(a) Find a solution v = f(x) to the differential equation satisfying f(0) =1/2
(b) Find the domain and range ofthe function f found in part (a).

Please see the attached file for the fully formatted problems.

#### Solution Summary

Derivatives and Differential Equations and a Leaking Tank Word Problem are investigated. The solution is detailed and well presented.

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