Midterm Problem Set Applications: First-Order Differential Equation

1.

A 200-gallon tank is half full of distilled water. At time t = 0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at the same rate. Find the amount X of concentrate in the tank after 30 minutes.

2.

A wet towel hung from a clothesline to dry loses moisture through evaporation at a rate proportional to its moisture content. If after 1 hour the towel has lost 40% of its original moisture content, after how long will it have lost 80%?

3.

Glucose is added intravenously to the bloodstream at the rate of q units per minute, and the body removes glucose from the bloodstream at a rate proportional to the amount present. Assume that Q(t) is the amount of glucose in the bloodstream at time t.

(a) Determine the differential equation describing the rate of change of glucose in the bloodstream with respect to time.

(b) Solve the differential equation from part (a), letting Q = Q0 when t = 0.

(c) Find the limit of Q(t) as t .

4.

A 200-gallon tank is full of solution containing 25 pounds of concentrate. Starting at time t = 0, distilled water is admitted to the tank at a rate of 10 gallons per minute, and the well-stirred solution is withdrawn at the same rate.

(a) Find the amount of concentrate X in the solution as a function of t.

(b) Find the time at which the amount of concentrate in the tank reaches 15 pounds.

(c) Find the quantity of the concentrate that the solution as t .

5.

A 200-gallon tank is half full of distilled water. At time t = 0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred mixture is withdrawn at the rate of 3 gallons per minute.

(a) After how many minutes will the tank be full?

(b) When the tank is full, how many pounds of concentrate will it contain?

6.

A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine containing pound of salt per gallon is pumped into the tank at a rate of 6 gal/min. The well-mixed solution is then pumped out at a slower rate of 4 gal/min. Find the number of pounds of salt in the tank after 30 minutes.

7.

A 30-volt electromotive force is applied to an LR-series circuit in which the inductance is 0.1 henry and the resistance is 50 ohms. Find the current i(t) if i(0) = 0. Determine the current as t .

8.

A 100-volt electromotive force is applied to an RC-series circuit in which the resistance is 200 ohms and the capacitance is 10-4 farad. Find the charge q(t) on the capacitor if q(0) = 0. Find the current i(t).

9. Refer to the circuit in Figure 1. LetvC(0) = 30 V. Determine vC, vx, and i0 for 0.

Figure 1

10. If the switch in Figure 2 opens at t = 0, find v(t) for 0 and (0).

Figure 2

11. Find i and vx in the circuit in Figure 3. Let i(0) = 5 A.

Figure 3

12. For the circuit of Figure 4 find i(t) for t > 0.

Figure 4

13. Find v(t) and v0(t) in the op amp circuit of Figure 5

Figure 5

14.

A nitric acid solution flows at a constant rate of 6 L/min into a large tank that initially held 200 L of a 0.5% nitric acid solution. The solution inside the tank is kept wells stirred and flows out of the tank at a rate of 8 L/min. If the solution entering the tank is 20% nitric acid, determine the volume of nitric acid in the tank after t min. When will the percentage of nitric acid in the tank reach 10%?

15.

A swimming pool whose volume is 10 000 gal contains water that is 0.01% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 5 gal/min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 hour? When will the pool water be 0.002% of chlorine?

Submission: ECE1A and ECE2A September 3CPE3 September 4

Rewrite (including diagrams, if applicable) the problem using Engineering Lettering in short bond paper. Use blue-inked pen only. Any form of alteration and/or erasure is not allowed. Enclose your final answer in a parallelogram. Use GRESA format in solving.

G Given

R Required

E Equation

S Solution

A - Answer

Sample

1. This is the problem from the handout.Given:Required:Equation:Solution:Answer: Final Answer