AP Calculus AB Section 7.3: Other Differential Equations for Real-World Applications Period: Date: Practice Exercises Score: / 5 Points

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1 AP Calculus AB Name: Section 7.3: Other Differential Equations for Real-World Applications Period: Date: Practice Exercises Score: / 5 Points 1. Sweepstakes Problem I: You have just won a national sweepstakes! Your award is an income of $100 a day for the rest of your life! You decide to put the money into a fireproof filing cabinet and let it accumulate. But temptation sets in, and you start spending the money at rate, in dollars per day. d. Suppose that each day you spend 2% of the money in your filing cabinet, so the proportionality constant in the equation for is Substitute this value into the equation you found in part c to get explicity in terms of. e. Plot the graph of versus, and sketch the result. a. Let be the number of dollars you have in the filing cabinet and be the number of days that you ve been receiving the money. Assuming that the rates are continuous, write a differential equation that expresses in terms of. b. Your spending rate,, is directly proportional to the amount of money,. Write an equation that expresses this fact, then substitute the result into the differential equation. f. How much money will you have in the filing cabinet after 30 days, 60 days, and 90 days? How much has come in? How much have you spent? c. Separate the variables and integrate the differential equation in part b to get an equation for in terms of. For the initial condition realize that when. g. How much money do you have in the filing cabinet after one year? At what rate is the amount increasing at this time? h. What is the limit of as approaches infinity?

2 2. Electrical Circuit Problem: When you turn on the switch in an electric circuit, a constant voltage (electrical pressure ),, is applied instantaneously to the circuit. This voltage causes an electrical current to flow through the circuit. The current is A (ampere) when the switch is turned on at time s. The part of this voltage that goes into overcoming the electrical resistance of the circuit is directly proportional to the current,. The proportionality constant,, is called the resistance of the circuit. The rest of the voltage is used to get the current moving through the circuit in the first place and varies directly with the instantaneous rate of change of the current with respect to time. The constant for this proportionality,, is called the inductance of the circuit. c. Suppose that the circuit has a resistance of 10 Ω (ohms) and an inductance of 20 H (henries). If the circuit is connected to a normal 110-V (volt) outlet, write the particular equation and plot the graph. Sketch the results and show any asymptotes. d. Predict the current for these times: i. 1 s after the switch is turned on a. Write a differential equation stating that is the sum of the resistive voltage and the inductive voltage. ii. 10 s after the switch is turned on iii. At a steady state, after many seconds b. Solve this differential equation subject to the initial condition that when. Write the resulting equation with as a function of. e. At what time,, will the current reach 95% of its steady-state value? 3. Newton s Law of Cooling Problem: When you turn on an electric heater, such as a burner on a stove, its temperature increases rapidly at first, then more slowly, and finally approaches a constant high temperature. As the burner warms up, heat supplied by the electricity goes to two places. i. Storage in the heater materials, thus warming the heater ii. Losses to the room

3 Assume that heat is supplied at a constant rate,. The rate at which heat is stored is directly proportional to the rate of change of temperature. Let be the number of degrees above room temperature. Let be the elapsed time, in seconds, since heat was applied. Then the storage rate is. The proportionality constant,, (calories per degree), is called the heat capacity of the heater materials. According to Newton s law of cooling, the rate at which heat is lost to the room is directly proportional to. The (positive) proportionality constant,, is called the heat transfer coefficient. a. The rate at which heat is supplied to the heater is equal to the sum of the storage rate and the loss rate. Write a differential equation that expresses this fact. d. Plot the graph of versus. Sketch the result. e. Predict at times of 10, 20, 50, 100, and 200 s after the heater is turned on. b. Separate the variables and integrate the differential equation. Transform the answer so that temperature,, is in terms of time,. Use the initial condition that when. f. Find the limit of as approaches infinity. This is called the steady-state temperature. g. How long does it take the heater to reach 99% of its steady-state temperature? c. Suppose that heat is supplied at a rate cal/s. Assume that the heat capacity is cal/deg and that the heat transfer coefficiet is (cal/s)/deg. Substitute these values to get in terms of alone. 4. Hot Tub Problem: The figure shows a cylindrical hot tub 8 ft in diameter and 4 ft deep. At time min, the drain is opened and water flows out. The rate at which it flows is proportional to the square root of the depth,, in feet. Because the tub has vertical sides, the rate is also proportional to the square root of the volume,, in cubic feet, of water left.

4 e. Does this mathematical model predict a time when the tub is completely empty, or does the volume,, approach zero asymptotically? If there is a time, state it. a. Write a differential equation for the rate at which water flows from the tub. That is, write an equation for in terms of. f. Draw a graph of versus in a suitable domain. b. Separate the variables and integrate the differential equation you wrote in part a. Transform the result so that is expressed explicity in terms of. Tell how varies with. c. Suppose that the tub intially (when ) contains 196 ft 3 of water and that when the drain is first opened the water flows out at 28 ft 3 /min (when, ). Find the particular solution of the differential equation that fits these initial conditions. 5. Differential Equation Generalization Problem: The solutions of are functions with different behaviors, depending on the value of the (constant) exponent. If, then varies exponentially with. If, as the Hot Tub Problem, then varies quadratically with. In this problem you will explore the graphs of variaous solutions of this equation. a. Write the solution of the equation for. Let and the constant of integration. Graph the solution and sketch the graph. d. Naïve thinking suggests that the tub will be empty after 7 min since it contains 196 ft 3 and the water is flowing out at 28 ft 3 /min. Show that this conclusion is false, and justify your answer.

5 b. Solve the equation for. Let and, as in part a. Graph the solution. d. Show that if then there is a vertical asymptote at. Plot two graphs that show the difference in behavior for and for. Use and, as in part a. c. Show that if, then is a square root function of, and if, then is a cube root function of. Plot both graphs, using and, as in part a. e. What type of function is when? Graph this function, using and, as in parts a-d.