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Jon Rogawski, Calculus: Early Transcendentals, W. H. Freeman and Company, 2008

Section 9.1

In Exercises 3–9, verify that the given function is a solution of the differential equation.
3. y″ + 8x = 0, y = 4x2

In Exercises 13–28, solve using separation of variables.
13. y′ = xy2

In Exercises 29–41, solve the initial value problem.
29. y′ + 2y = 0, y(ln(2)) = 3

45. A cylindrical tank filled with water has height 10 ft and a base of area 30 ft2. Water leaks through a hole in the bottom of area 1/3 ft2. How long does it take (a) for half of the water to leak out and (b) for the tank to empty?

46. A conical tank filled with water has height 12 ft [Figure 7(A)]. Assume that the top is a circle of radius 4 ft and that water leaks through a hole in the bottom of area 2 in2. Let y ( t ) be the water level at time t. (a) Show that the cross-sectional area of the tank at height y is A ( y ) = (π/9)y2. (b) Find the differential equation satisfied by y ( t ) and solve for y ( t ). Use the initial condition y(0) = 12. (c) How long does it take for the tank to empty?

47. The tank in Figure 7(B) is a cylinder of radius 10 ft and length 40 ft. Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area B = 3 in2. Determine the water level y(t) and the time te when the tank is empty.

48. A cylindrical tank filled with water has height h ft and a base of area A ft2. Water leaks through a hole in the bottom of area B ft2. (a) Show that the time required for the tank to empty is proportional to A√h/B. (b) Show that the emptying time is proportional to Vh−1/2, where V is the volume of the tank. (c) Two tanks have the same volume and same-sized hole, but different heights and bases. Which tank empties first: the taller or shorter?

49. Figure 8 shows a circuit consisting of a resistor of R ohms, a capacitor of C farads, and a battery of voltage V. When the circuit is completed, the amount of charge q (t) (in coulombs) on the plates of the capacitor varies according to the differential equation (t in seconds) R dq/dt + q/C = V. (a) Solve for q(t). (b) Show that lim(t→∞) q(t) = CV (c) Find q ( t ) , assuming that q(0) = 0. Show that the capacitor charges to approximately 63% of its final value CV after a time period of length τ= RC (τ is called the time constant of the capacitor).

50. Assume in the circuit of Figure 8 that R = 100 Ω , C = 0.01 F, and V = 10 V. How many seconds does it take for the charge on the capacitor plates to reach half of its limiting value?

52. We might also guess that the rate at which a snowball melts is proportional to its surface area. What is the differential equation satisfied by the volume V of a spherical snowball at time t? Suppose the snowball has radius 4 cm and that it loses half of its volume after 10 min. According to this model, when will the snowball disappear?

54. A boy standing at point B on a dock holds a rope of length l attached to a boat at point A [Figure 9(A)]. As the boy walks along the dock, holding the rope taut, the boat moves along a curve called a tractrix (from the Latin tractus meaning “to pull”). The segment from a point P on the curve to the x-axis along the tangent line has constant length l. Let y = f (x) be the equation of the tractrix. (a) Show that y2 + (y/y′)2 = l2 and conclude y′ = -y/√(l2 – y2). Why must we choose the negative square root? (b) Prove that the tractrix is the graph of x = lln((l + √(l2 – y2))/y) - √(l2 – y2).

55. If a bucket of water spins about a vertical axis with constant angular velocity ω (in radians per second), the water climbs up the side of the bucket until it reaches an equilibrium position (Figure 10). Two forces act on a particle located at a distance x from the vertical axis: the gravitational force − mg acting downward and the force of the bucket on the particle (transmitted indirectly through the liquid) in the direction perpendicular to the surface of the water. These two forces must combine to supply a centripetal force mω2x, and this occurs if the diagonal of the rectangle in Figure 10 is normal to the water’s surface (that is, perpendicular to the tangent line). Prove that if y = f (x) is the equation of the curve obtained by taking a vertical cross section.

58. A 50-kg model rocket lifts off by expelling fuel at a rate of k = 4.75 kg/s for 10 s. The fuel leaves the end of the rocket with an exhaust velocity of b = 100 m/s. Let m (t) be the mass of the rocket at time t. From the law of conservation of momentum, we find the following differential equation for the rocket’s velocity v(t) (in meters per second): m(t)v′( t ) = − 9.8m(t) + b dm/dt. (a) Show that m(t) = 50− 4.75t kg. (b) Solve for v(t) and compute the rocket’s velocity at rocket burnout (after 10 s).

59. Let v(t) be the velocity of an object of mass m in free fall near the earth’s surface. If we assume that air resistance is proportional to v2, then v satisfies the differential equation m dv/dt = − g + kv2 for some constant k > 0. (a) Set α = (g/k)1/2 and rewrite the differential equation as dv/dt = −k/m(α2 − v2). Then solve using separation of variables with initial condition v(0) = 0. (b) Show that the terminal velocity lim(t →∞) v( t ) is equal to −α.


S
ection 9.2


1. Find the general solution of y ′ = 2(y − 10). Then find the two solutions satisfying y(0) = 25 and y(0) = 5, and sketch their graphs.

4. Let F(t) be the temperature of a hot object submerged in a large pool of water whose temperature is 70 °F. (a) What is the differential equation satisfied by F(t) if the cooling constant is k = 1.5? (b) Find a formula for F(t) if the object’s initial temperature is 250 °F.

5. A hot metal bar is submerged in a large reservoir of water whose temperature is 60 °F. The temperature of the bar 20 s after submersion is 100 °F. After 1 min, the temperature has cooled to 80 °F. (a) Determine the cooling constant k. (b) What is the differential equation satisfied by the temperature F(t) of the bar? (c) What is the formula for F(t)? (d) Determine the temperature of the bar at the moment it is submerged.

6. A hot metal rod is placed in a water bath whose temperature is 40 °F. The rod cools from 300 to 200 °F in 1 min. How long will it take for the rod to cool to 150 °F?

7. When a hot object is placed in a water bath whose temperature is 25 °C, it cools from 100 to 50 °C in 150 s. In another bath, the same cooling occurs in 120 s. Find the temperature of the second bath.

8. A cold metal bar at −30 °C is submerged in a pool maintained at a temperature of 40 °C. Half a minute later, the temperature of the bar is 20 °C. How long will it take for the bar to attain a temperature of 30 °C?

9. A cup of coffee, cooling off in a room at temperature 20 °C, has cooling constant k = 0.09 min−1.
(a) How fast is the coffee cooling (in degrees per minute) when its temperature is T = 80 °C? (b) Use the Linear Approximation to estimate the change in temperature over the next 6 s when T = 80 °C. (c) The coffee is served at a temperature of 90 °C. How long should you wait before drinking it if the optimal temperature is 65 °C?

10. Two identical objects are heated to different temperatures T1 and T2. Both are submerged in a cold bath of temperature T0 = 40 °C at t = 0. Measurements show that T2 = 400 °C and the cooling rate of object 1 is twice as large as the cooling rate of object 2 (at each time t). Find T1.

11. A 60-kg skydiver jumps out of an airplane. What is her terminal velocity in miles per hour, assuming that k = 10 kg /s for free-fall (no parachute)?

12. Find the terminal velocity of a 192-lb skydiver if k = 1.2 lb-s/ft. How long does it take him to reach half of his terminal velocity if his initial velocity is zero?

13. A 175-lb skydiver jumps out of an airplane (with zero initial velocity). Assume that k = 0.7 lb-s/ft with a closed parachute and k = 5 lb-s/ft with an open parachute. What is the skydiver’s velocity at t = 25 s if the parachute opens after 20 seconds of free fall?

14. Does a heavier or lighter skydiver reach terminal velocity faster?

15. A continuous annuity with withdrawal rate N = $1000/year and interest rate r = 5% is funded by an initial deposit P0. (a) When will the annuity run out of funds if P0 = $15000? (b) Which initial deposit P0 yields a constant balance?

16. Show that a continuous annuity with withdrawal rate N = $5000/year and interest rate r = 8%, funded by an initial deposit of P0 = $75000, never runs out of money.

17. Find the minimum initial deposit that will allow an annuity to pay out $500 / year indefinitely if it earns interest at a rate of 5%.

18. What is the minimum initial deposit necessary to fund an annuity for 30 years if withdrawals are made at a rate of $2000 / year at an interest rate of 7%?

19. An initial deposit of $5,000 is placed in a bank account. What is the minimum interest rate the bank must pay to allow continuous withdrawals at a rate of $500/year to continue indefinitely?

20. Show that a continuous annuity never runs out of money if the initial balance is greater than or equal to N/r, where N is the withdrawal rate and r the interest rate.

21. Sam borrows $10,000 from a bank at an interest rate of 9% and pays back the loan continuously at a rate of N dollars/year. Let P (t) denote the amount still owed at time t. (a) Explain why P(t) satisfies the differential equation y′ = 0.09y – N (b) How long will it take Sam to pay back the loan if N = $1200? (c) Will he ever be able to pay back the loan if N = $800?

22. Let N (t) be the fraction of the population who have heard a given piece of news t hours after its initial release. According to one model, the rate N′( t ) at which the news spreads is equal to k times the fraction of the population that has not yet heard the news, for some constant k. (a) Determine the differential equation satisfied by N(t). (b) Find the solution of this differential equation with the initial condition N(0) = 0 in terms of k. (c) Suppose that half of the population is aware of an earthquake 8 hours after it occurs. Use the model to calculate k and estimate the percentage that will know about the earthquake 12 hours after it occurs.

23. Current in a Circuit. The electric current flowing in the circuit in Figure 6 (consisting of a battery of V volts, a resistor of R ohms, and an inductor) satisfies dI/dt = − k(I − b) for some constants k and b with k > 0. Initially, I(0) = 0 and I(t) approaches a maximum level V/R as t → ∞.

24. Show that the cooling constant of an object can be determined from two temperature readings y ( t1) and y(t2) at times t1  = t2 by the formula k = 1/(t1- t2) ln((y(t2) –T0)/(y(t1) –T0))

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