A pitcher of buttermilk initially at 25°C is to be cooled

Problem 3P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 69P Suppose a uniform flexible cable is suspended between two points (±L, H) at equal heights located symmetrically on either side of the x ? axis. Principles of physics can be used to show that the shape y = y(x) of Lhe hanging cable satisfies the differential equation0 where the constant a = T/?; is the ratio of the cable’s tension T at its lowest point x = 0 (where y?(0) = 0) and its (constant) linear density ?. If we substitute u = dymyslashdx, dv/dx = d2y/dx2 in this second-order differential equation, we get the first-order equation Solve this differential equation for y?(x) = v(x) = sinh(x/a). Then integrate to get the shape function of the hanging cable. This curve is called a catenaiy, from the Latin word for chain. FIGURE. The catenary.

Problem 4P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 2P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 1P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 5P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 7P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 6P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 9P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 8P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 10P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 12P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 13P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 11P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 15P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 14P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 16P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 17P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x. (suggestion: Factor the right-hand side.)

Problem 18P Find general solutions (implicit if necessary, explicit if convenient) of the differential equations in Problems. Primes denote derivatives with respect to x.

Problem 21P Find explicit particular solutions of the initial value problems In Problems.

Problem 20P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 19P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 22P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 23P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 24P Find explicit particular solutions of the initial value problems in Problems.

Problem 25P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 28P Find explicit particular solutions of the initial value problems In Problems.

Problem 26P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 29P (a) Find a general solution of the differential equation dy/dx = y2. (b) Find a singular solution that is not included in the general solution, (c) Inspect a sketch of typical solution curves to determine the points (a, b) for which the initial value problem y? = y2, y(a) = b has a unique solution.

Problem 27P Find explicit particular solutions of the initial value problemsIn Problems.

Problem 30P Solve the differential equation (dy/dx)2 = 4y to verify the general solution curves and singular solution curve that are illustrated in Fig. Then determine the points (a, b) in the plane for which the initial value problem (y?)2 = 4y, y(a) = b has (a) no solution, (b) infinitely many solutions that are defined for all x, (c) on some neighborhood of the point x = a, only finitely many solutions.

Problem 31P Discuss the difference between the differential equations Do they have the same solution curves? Why or why not? Determine the points (a, b) in the plane for which the initial value problem , y(a) = b has (a) no solution, (b) a unique solution, (c) infinitely many solutions.

Problem 33P (Population growth) A certain city had a population of 25000 in 1960 and a population of 30000 in 1970. Assume that its population will continue to grow exponentially at a constant rate. What population can its city planners expect in the year 2000?

Problem 32P Find a general solution and any singular solutions of the differential equation dymyslash Determine the points (a, b) in the plane tor which the initial value problem has (a) no solution, (b) a unique solution, (c) infinitely many solutions.

Problem 34P (Population growth) In a certain culture of bacteria, the number of bacteria increased sixfold in 10 h. How long did it take for the population to double?

Problem 36P (Radiocarbon dating) Carbon taken from a purported relic of the time of Christ contained 4.6 × 1010 atoms of 14C per gram. Carbon extracted from a present-day specimen of the same substance contained 5.0 × 1010 atoms of 14C per gram. Compute the approximate age of the relic. What is your opinion as to its authenticity?

Problem 35P (Radiocarbon dating) Carbon extracted from an ancient skull contained only one-sixth as much 14C as carbon extracted from present-day bone. How old is the skull?

Problem 37P (Continuously compounded interest) Upon the birth of their first child, a couple deposited $5000 in an account that pays 8% interest compounded continuously. The interest payments are allowed to accumulate. How much will the account contain on the child’s eighteenth birthday?

Problem 38P (Continuously compounded interest) Suppose that you discover in your attic an overdue library book on which your grandfather owed a fine of 30 cents 100 years ago. If an overdue fine grows exponentially at a 5% annual rate compounded continuously, how much would you have to pay if you returned the book today?

Problem 40P The half-life of radioactive cobalt is 5.27 years. Suppose that a nuclear accident has left the level of cobalt radiation in a certain region at 100 times the level acceptable for human habitation. How long will it be until the region is again habitable? (Ignore the probable presence of other radioactive isotopes.)

Problem 39P (Drug elimination) Suppose that sodium pentobarbital is used to anesthetize a dog. The dog is anesthetized when its bloodstream contains at least 45 milligrams (mg) of sodium pentobarbitol per kilogram of the dog’s body weight. Suppose also that sodium pentobarbital is eliminated exponentially from the dog’s bloodstream, with a half-life of 5 h. What single dose should be administered in order to anesthetize a 50-kg dog for 1 h?

Problem 41P Suppose that a mineral body formed in an ancient cataclysm-perhaps the formation of the earth itself-originally contained the uranium isotope 238U (which has a half-life of 4.51 x 109 years) but no lead, the end product of the radioactive decay of 238U. If today the ratio of 238U atoms to lead atoms in the mineral body is 0.9, when did the cataclysm occur?

Problem 43P A pitcher of buttermilk initially at 25°C is to be cooled by setting it on the front porch, where the temperature is 0°C. Suppose that the temperature of the buttermilk has dropped to 15°C after 20 min. When will it be at 5°C?

Problem 42P A certain moon rock was found to contain equal numbers of potassium and argon atoms. Assume that all the argon is the result of radioactive decay of potassium (its half-life is about 1.28 x 109 years) and that one of every nine potassium atom disintegrations yields an argon atom. What is the age of the rock, measured from the time it contained only potassium?

Problem 44P When sugar is dissolved in water, the amount A that remains undissolved after t minutes satisfies the differential equation dA/dt = ?kA (k > 0). If 25% of the sugar dissolves after 1 min, how long does it take for half of the sugar to dissolve?

Problem 45P The intensity I of light at a depth of x meters below the surface of a lake satisfies the differential equation dI/dx = (?1.4) I. (a) At what depth is the intensity half the intensity I0 at the surface (where x = 0)? (b) What is the intensity at a depth of 10 m (as a fraction of I0)? (c) At what depth will the intensity be 1% of that at the surface?

Problem 46P The barometric pressure p (in inches of mercury) at an altitude x miles above sea level satisfies the initial value problem dp/dx = (?0.2)p, p(0) = 29.92. (a) Calculate the barometric pressure at 10,000 ft and again at 30,000 ft. (b) Without prior conditioning, few people can survive when the pressure drops to less than 15 in. of mercury. How high is that?

Problem 47P A certain piece of dubious information about phenylethy-larninc in the drinking water began to spread one day in a city with a population of 100,000. Within a week, 10,000 people had heard this rumor. Assume that the rate of increase of the number who have heard the rumor is proportional to the number who have not yet heard it. How long will it be until half the population of the city has heard the rumor?

Problem 49P A cake is removed from an oven at 210°F and left to cool at room temperature, which is 70° F. After 30 min the temperature of the cake is 140°F. When will it be 100°F?

Problem 48P According to one cosmological theory, there were equal amounts of the two uranium isotopes 235U and 238U at the creation of the universe in the “big bang.” At present there are 137.7 atoms of 238U for each atom of 235U. Using the half-lives 4.51 x 109 years for 238U and 7.10 x 108 years for 235U, calculate the age of the universe.

The amount \(A(t)\) of atmospheric pollutants in a certain mountain valley grows naturally and is tripling every \(7.5\) years. (a) If the initial amount is \(10 \mathrm{pu}\) (pollutant units), write a formula for \(A(t)\) giving the amount (in pu) present after \(t\) years. (b) What will be the amount (in pu) of pollutants present in the valley atmosphere after 5 years? (c) If it will be dangerous to stay in the valley when the amount of pollutants reaches \(100 \mathrm{pu}\), how long will this take?

Problem 51P An accident at a nuclear power plant has left the surrounding area polluted with radioactive material that decays naturally. The initial amount of radioactive material present is 15 sii (safe units), and 5 months later it is still 10 su. (a) Write a formula giving the amount A(t) of radioactive material (in su) remaining after t months. ________________ (b) What amount of radioactive material will remain after 8 months? ________________ (c) How long-total number of months or fraction thereof-will it be until A = 1 su, so it is safe for people to return to the area?

Problem 52P There are now about 3300 different human “language families” in the whole world. Assume that all these are derived from a single original language, and that a language family develops into 1.5 language families every 6 thousand years. About how long ago was the single original human language spoken?

Problem 53P Thousands of years ago ancestors of the Native Americans crossed the Bering Strait from Asia and entered the western hemisphere. Since then, they have fanned out across North and South America. The single language that the original Native Americans spoke has since split into many Indian “language families.” Assume (as in Problem) that the number of these language families has been multiplied by 1.5 every 6000 years. There are now 150 Native American language families in the western hemisphere. About when did the ancestors of today’s Native Americans arrive? Problem Thousands of years ago ancestors of the Native Americans crossed the Bering Strait from Asia and entered the western hemisphere. Since then, they have fanned out across North and South America. The single language that the original Native Americans spoke has since split into many Indian “language families.” Assume (as in Problem) that the number of these language families has been multiplied by 1.5 every 6000 years. There are now 150 Native American language families in the western hemisphere. About when did the ancestors of today’s Native Americans arrive?

Problem 54P A tank is shaped like a vertical cylinder; it initially contains water to a depth of 9 ft, and a bottom plug is removed at time t = 0 (hours). After 1 h the depth of the water has dropped to 4 ft. How long does it take for all the water to drain from the tank?

Problem 56P At time t = 0 the bottom plug (at the vertex) of a full conical water tank 16 ft high is removed. After I h the water in the tank is 9 ft deep. When will the tank be empty?

Problem 57P Suppose that a cylindrical tank initially containing Vo gallons of water drains (through a bottom hole) in T minutes. Use Torricelli’s law to show that the volume of water in the tank after t ? Tminutes is V = Vo [1 ? (t/T)]2.

Problem 58P A water tank has the shape obtained by revolving the curve y = x4/3 around the y ? axis. A plug at the bottom is removed at 12 noon, when the depth of water in the tank is 12 ft. At 1 P.M. the depth of the water is 6 ft. When will the tank be empty?

Problem 55P Suppose that the tank of Problem has a radius of 3 ft and that its bottom hole is circular with radius 1 in. How long will it take the water (initially 9 ft deep) to drain completely? Reference Problem: A tank is shaped like a vertical cylinder; it initially contains water to a depth of 9 ft, and a bottom plug is removed at time t= 0 (hours). After 1 h the depth of the water has dropped to 4 ft. How long does it take for all the water to drain from the tank?

Problem 60P A cylindrical tank with length 5 ft and radius 3 ft is situated with its axis horizontal. If a circular bottom hole with a radius of 1 in. is opened and the tank is initially half full of xylene, how long will it take for the liquid to drain completely?

Problem 59P A water tank has the shape obtained by revolving the parabola x2 = by around the y ? axis. The water depth is 4 ft at 12 noon, when a circular plug in the bottom of the tank is removed. At 1 P.M. the depth of the water is 1 ft. (a) Find the depth y(t) of water remaining after t hours. ________________ (b) When will the tank be empty? (c) if the initial radius of the top surface of the water is 2 ft, what is the radius of the circular hole in the bottom?

Problem 61P A spherical tank of radius 4 ft is full of gasoline when a circular bottom hole with radius 1 in. is opened. How long will be required for all the gasoline to drain from the tank?

Problem 62P Suppose that an initially full hemispherical water tank of radius 1 m has its flat side as its bottom. It has a bottom hole of radius 1 cm. If this bottom hole is opened at 1 P.M., when will the tank be empty?

Problem 63P Consider the initially full hemispherical water tank of Example 8, except that the radius r of its circular bottom hole is now unknown. At 1 P.M. the bottom hole is opened and at 1:30 P.M. the deoth of water in the tank is 2 ft. (a) Use Torricelli’s law in the form (taking constriction into account) to determine when the tank will be empty, (b) What is the radius of the bottom hole?

Problem 65P Just before midday the body of an apparent homicide victim is found in a room that is kept at a constant temperature of 70°F. At 12 noon the temperature of the body is 80°F and at 1 P.M. it is 75°F. Assume that the temperature of the body at the time of death was 98.6°F and that it has cooled in accord with Newton’s law. What was the time of death?

Problem 66P Early one morning it began to snow at a constant rate. At 7 A.M. a snowplow set off to clear a road. By 8 A.M. it had traveled 2 miles, but it took two more hours (until 10 A.M.) for the snowplow to go an additional 2 miles, (a) Let t = 0 when it began to snow and let x denote the distance traveled by the snowplow at time t. Assuming that the snowplow clears snow from the road at a constant rate (in cubic feet per hour, say), show that where k is a constant, (b) What time did it start snowing? (Answer: 6 A.M.)

Problem 67P A snowplow sets off at 7 A.M. as in Problem 66. Suppose now that by 8 A.M. it had traveled 4 miles and that by 9 A.M. it had moved an additional 3 miles. What time did it start snowing? This is a more difficult snowplow problem because now a transcendental equation must be solved numerically to find the value of k. (Answer: 4:27 A.M.)

Problem 68P shows a bead sliding down a frictionless wire from point P to point Q. The brachistochrone problem asks what shape the wire should be in order to minimize the bead’s time of descent from P to Q. In June of 1696, John Bernoulli proposed this problem as a public challenge, with a 6-month deadline (later extended to Easter 1697 at George Leibniz’s request). Isaac Newton, then retired from academic life and serving as Warden of the Mint in London, received Bernoulli’s challenge on January 29, 1697. The very next day he communicated his own solutionof the tangent line to the curve–the curve of minimal descent time is an are of an inverted cycloid–to the Royal Society of London. For a modern derivation of this result, suppose the bead starts from rest at the origin P and let y = y(x) be the equation of the desired curve in a coordinate system with the y ? axis pointing downward. Then a mechanical analogue of Snell’s law in optics implies that where ? denotes the angle of deflection (from the vertical) of the tangent line to the curve–so cot? = y?(x) (why?)–and is the bead’s velocity when it has descended a distance yvertically (from KE = ½mv2 = mgy = ?PE). FIGURE. A bead sliding down a wire-the brachistochrone problem. (a) First derive from Eq. (i) the differential equation where a is an appropriate positive constant. ________________ (b) Substitute y = 2a sin2 t, dy = 4a sin t cos t dt in (ii) to derive the solution x = a(2t ? sin 2t), y = a (1 ? cos 2t) (iii) for which t = y = 0 when x = 0. Finally, the substitution of ? = 2a in (iii) yields the standard parametric equations x = a(? ? sin?), y = a(1 ? cost?) of the cycloid that is generated by a point on the rim of a circular wheel of radius a as it rolls along the x-axis. [See Example 5 in Section 9.4 of Edwards and Penney, Calculus: Early Transcendentals, 7th edition (Upper Saddle River, NJ: Prentice Hall, 2008).]

Problem 64P (The clepsydra, or water clock) A 12-h water clock is to be designed with the dimensions shown in Fig, shaped like the surface obtained by revolving the curve y = f(x) around the y ? axis. What should be this curve, and what should be the radius of the circular bottom hole, in order that the water level will fall at the constant rate of 4 inches per hour (in./h)? FIGURE The clepsydra.