# 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 √(t+1) gallons per minute for 0 ≤ r ≤ 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 ≤ t ≤ 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).

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Solution:

Question 1: water tank problem

a) From t=0 to t=3 min, water leakage is:

b) At t=3 min, we know the leakage and the pumped in water is: 3*8=24 gallon, then at t=3min, ...

#### Solution Summary

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