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Math Advanced Math Fundamentals of Differential Equations plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (9th Edition) (Nagle, Saff & Snider, Fundamentals of Differential Equations)

The constant temperature of the kitchen by the method of Newton’s law of cooling. The constant temperature of the kitchen is 20 ° C _ . Given: The water temperature in a pot is 100 ° C at the time of removed from a stove. Take t = 0 when the water temperature in a pot is 100 ° C . The water temperature has decreased to 80 ° C after 5 minutes. The water temperature has decreased to 65 ° C after 10 minutes. Calculation: Let T be the water temperature and M be the constant temperature of the kitchen. The Problem 34 states that, “According to Newton’s law of cooling, if an object at temperature T is immersed in a medium having the constant temperature M then the rate of change of T is proportional to the difference of temperature M − T This gives the differential equation is, d T d t = k ( M − T ) ”. Assume Newton’s law of cooling (Problem 34) applies in this problem. Separate the variables and rewrite the differential equation as follows. d T d t = k ( M − T ) 1 ( M − T ) d T = k d t Integrate on both sides of the equation. ∫ 1 ( M − T ) d T = ∫ k d t − ln | M − T | = k t + C 1 ln | ( M − T ) − 1 | = k t + C 1 ln | 1 M − T | = k t + C 1 Take exponential on both side of the equation. e ln ( 1 M − T ) = e k t + C 1 1 M − T = e C 1 e k t 1 M − T = C 2 e k t M − T = C 3 e − k t Simplify further as follows. − T = − M + C 3 e − k t T = M − C 3 e − k t T = M + C e − k t ( − C 3 = C , where C is an arbitrary constant ) T = M + C e − k t ... ( 1 ) Now find the constant temperature of the kitchen M by using the above equation (1). Since the water temperature is 100 ° C at t = 0 , substitute t = 0 and T = 100 in the above solution. T = M + C e − k t 100 = M + C e − k ( 0 ) 100 = M + C C = 100 − M ... ( 2 ) Since the water temperature has decreased to 80 ° C after 5 minutes, substitute t = 5 and T = 80 in equation (1). 80 = M + C e − k ( 5 ) 80 = M + C e − 5 k e − 5 k = 80 − M C ( e − 5 k ) 2 = ( 80 − M C ) 2 ( on squaring both sides ) e − 10 k = ( 80 − M ) 2 C 2 ... ( 3 ) Since the water temperature has decreased to 65 ° C after 10 minutes, substitute t = 10 and T = 65 in equation (1). T = M + C e − k t 65 = M + C e − 10 k C e − 10 k = 65 − M e − 10 k = 65 − M C ... ( 4 ) Equate the equations (3) and (4). ( 80 − M ) 2 C 2 = 65 − M C ( 80 − M ) 2 65 − M = C 2 C ( 80 − M ) 2 65 − M = C Substitute C = 100 − M in the above result. ( 80 − M ) 2 ( 65 − M ) = ( 100 − M ) ( 80 − M ) 2 = ( 100 − M ) ( 65 − M ) 6400 − 160 M + M 2 = 6500 − 100 M − 65 M + M 2 − 160 M + 100 M + 65 M = 6500 − 6400 + M 2 − M 2 Simplify further as follows. 5 M = 100 M = 100 5 M = 20 ° C Thus, the constant temperature of the kitchen is 20 ° C _ .

The constant temperature of the kitchen by the method of Newton’s law of cooling. The constant temperature of the kitchen is 20 ° C _ . Given: The water temperature in a pot is 100 ° C at the time of removed from a stove. Take t = 0 when the water temperature in a pot is 100 ° C . The water temperature has decreased to 80 ° C after 5 minutes. The water temperature has decreased to 65 ° C after 10 minutes. Calculation: Let T be the water temperature and M be the constant temperature of the kitchen. The Problem 34 states that, “According to Newton’s law of cooling, if an object at temperature T is immersed in a medium having the constant temperature M then the rate of change of T is proportional to the difference of temperature M − T This gives the differential equation is, d T d t = k ( M − T ) ”. Assume Newton’s law of cooling (Problem 34) applies in this problem. Separate the variables and rewrite the differential equation as follows. d T d t = k ( M − T ) 1 ( M − T ) d T = k d t Integrate on both sides of the equation. ∫ 1 ( M − T ) d T = ∫ k d t − ln | M − T | = k t + C 1 ln | ( M − T ) − 1 | = k t + C 1 ln | 1 M − T | = k t + C 1 Take exponential on both side of the equation. e ln ( 1 M − T ) = e k t + C 1 1 M − T = e C 1 e k t 1 M − T = C 2 e k t M − T = C 3 e − k t Simplify further as follows. − T = − M + C 3 e − k t T = M − C 3 e − k t T = M + C e − k t ( − C 3 = C , where C is an arbitrary constant ) T = M + C e − k t ... ( 1 ) Now find the constant temperature of the kitchen M by using the above equation (1). Since the water temperature is 100 ° C at t = 0 , substitute t = 0 and T = 100 in the above solution. T = M + C e − k t 100 = M + C e − k ( 0 ) 100 = M + C C = 100 − M ... ( 2 ) Since the water temperature has decreased to 80 ° C after 5 minutes, substitute t = 5 and T = 80 in equation (1). 80 = M + C e − k ( 5 ) 80 = M + C e − 5 k e − 5 k = 80 − M C ( e − 5 k ) 2 = ( 80 − M C ) 2 ( on squaring both sides ) e − 10 k = ( 80 − M ) 2 C 2 ... ( 3 ) Since the water temperature has decreased to 65 ° C after 10 minutes, substitute t = 10 and T = 65 in equation (1). T = M + C e − k t 65 = M + C e − 10 k C e − 10 k = 65 − M e − 10 k = 65 − M C ... ( 4 ) Equate the equations (3) and (4). ( 80 − M ) 2 C 2 = 65 − M C ( 80 − M ) 2 65 − M = C 2 C ( 80 − M ) 2 65 − M = C Substitute C = 100 − M in the above result. ( 80 − M ) 2 ( 65 − M ) = ( 100 − M ) ( 80 − M ) 2 = ( 100 − M ) ( 65 − M ) 6400 − 160 M + M 2 = 6500 − 100 M − 65 M + M 2 − 160 M + 100 M + 65 M = 6500 − 6400 + M 2 − M 2 Simplify further as follows. 5 M = 100 M = 100 5 M = 20 ° C Thus, the constant temperature of the kitchen is 20 ° C _ .

Solution Summary: The author calculates the constant temperature of the kitchen by the method of Newton's law of cooling.

Fundamentals of Differential Equations plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (9th Edition) (Nagle, Saff & Snider, Fundamentals of Differential Equations)

Fundamentals of Differential Equations plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (9th Edition) (Nagle, Saff & Snider, Fundamentals of Differential Equations)

9th Edition

ISBN: 9780134768748

Author: Nagle

Publisher: PEARSON

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Fundamentals of Differential Equations plus MyLab Math with Pearson eText -- Title-Specific Access Card Package (9th Edition) (Nagle, Saff & Snider, Fundamentals of Differential Equations)

Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 16, determine whether the given...Ch. 2.2 - In Problems 716, solve the equation. 7. xdydx=1y3 Ch. 2.2 - In Problems 716, solve the equation. 8. dxdt=3xt2 Ch. 2.2 - In Problems 716, solve the equation. 9....Ch. 2.2 - In Problems 716, solve the equation. 10....

Ch. 2.2 - In Problems 716, solve the equation. 11....Ch. 2.2 - In Problems 716, solve the equation. 12....Ch. 2.2 - In Problems 716, solve the equation. 13....Ch. 2.2 - In Problems 716, solve the equation. 14. dxdtx3=x Ch. 2.2 - In Problems 716, solve the equation. 15....Ch. 2.2 - In Problems 716, solve the equation. 16. y1 dy +...Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - Prob. 23E Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - In Problems 1726, solve the initial value problem....Ch. 2.2 - Prob. 27E Ch. 2.2 - Sketch the solution to the initial value problem...Ch. 2.2 - Uniqueness Questions. In Chapter 1 we indicated...Ch. 2.2 - As stated in this section, the separation of...Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 32E Ch. 2.2 - Mixing. Suppose a brine containing 0.3 kilogram...Ch. 2.2 - Newtons Law of Cooling. According to Newtons law...Ch. 2.2 - Prob. 35E Ch. 2.2 - Prob. 36E Ch. 2.2 - Compound Interest. If P(t) is the amount of...Ch. 2.2 - Free Fall. In Section 2.1, we discussed a model...Ch. 2.2 - Grand Prix Race. Driver A had been leading...Ch. 2.2 - Prob. 40E Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 16, determine whether the given...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - In Problems 716, obtain the general solution to...Ch. 2.3 - Prob. 15E Ch. 2.3 - Prob. 17E Ch. 2.3 - Prob. 18E Ch. 2.3 - Prob. 19E Ch. 2.3 - Prob. 20E Ch. 2.3 - In Problems 1722, solve the initial value problem....Ch. 2.3 - In Problems 1722, solve the initial value problem....Ch. 2.3 - Radioactive Decay. In Example 2 assume that the...Ch. 2.3 - Prob. 24E Ch. 2.3 - (a) Using definite integration, show that the...Ch. 2.3 - Prob. 26E Ch. 2.3 - Constant Multiples of Solutions. (a) Show that y =...Ch. 2.3 - Prob. 29E Ch. 2.3 - Bernoulli Equations. The equation (18) dydx+2y=xy2...Ch. 2.3 - Prob. 31E Ch. 2.3 - Prob. 32E Ch. 2.3 - Prob. 33E Ch. 2.3 - Prob. 34E Ch. 2.3 - Prob. 35E Ch. 2.3 - Prob. 36E Ch. 2.3 - Prob. 37E Ch. 2.3 - Prob. 38E Ch. 2.3 - Prob. 39E Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - Prob. 3E Ch. 2.4 - Prob. 4E Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - In Problems 18, classify the equation as...Ch. 2.4 - Prob. 9E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 11E Ch. 2.4 - Prob. 12E Ch. 2.4 - Prob. 13E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 15E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 17E Ch. 2.4 - In Problems 920, determine whether the equation is...Ch. 2.4 - Prob. 19E Ch. 2.4 - Prob. 20E Ch. 2.4 - In Problems 2126, solve the initial value problem....Ch. 2.4 - Prob. 22E Ch. 2.4 - Prob. 23E Ch. 2.4 - In Problems 2126, solve the initial value problem....Ch. 2.4 - Prob. 25E Ch. 2.4 - In Problems 2126, solve the initial value problem....Ch. 2.4 - Prob. 27E Ch. 2.4 - For each of the following equations, find the most...Ch. 2.4 - Prob. 29E Ch. 2.4 - Prob. 30E Ch. 2.4 - Prob. 31E Ch. 2.4 - Orthogonal Trajectories. A geometric problem...Ch. 2.4 - Prob. 33E Ch. 2.4 - Prob. 34E Ch. 2.4 - Prob. 35E Ch. 2.4 - Prob. 36E Ch. 2.5 - Prob. 1E Ch. 2.5 - In Problems 16, identify the equation as...Ch. 2.5 - Prob. 3E Ch. 2.5 - Prob. 4E Ch. 2.5 - In Problems 16, identify the equation as...Ch. 2.5 - Prob. 6E Ch. 2.5 - Prob. 7E Ch. 2.5 - Prob. 8E Ch. 2.5 - Prob. 9E Ch. 2.5 - Prob. 10E Ch. 2.5 - Prob. 11E Ch. 2.5 - Prob. 12E Ch. 2.5 - Prob. 13E Ch. 2.5 - Prob. 14E Ch. 2.5 - Prob. 15E Ch. 2.5 - Prob. 16E Ch. 2.5 - Prob. 17E Ch. 2.5 - Prob. 18E Ch. 2.5 - Prob. 19E Ch. 2.5 - Verify that when the linear differential equation...Ch. 2.6 - In Problems 18, identify (do not solve) the...Ch. 2.6 - Prob. 2E Ch. 2.6 - Prob. 3E Ch. 2.6 - Prob. 4E Ch. 2.6 - Prob. 5E Ch. 2.6 - Prob. 6E Ch. 2.6 - In Problems 18, identify (do not solve) the...Ch. 2.6 - Prob. 8E Ch. 2.6 - Use the method discussed under Homogeneous...Ch. 2.6 - Prob. 10E Ch. 2.6 - Prob. 11E Ch. 2.6 - Prob. 12E Ch. 2.6 - Prob. 13E Ch. 2.6 - Prob. 14E Ch. 2.6 - Prob. 15E Ch. 2.6 - Prob. 16E Ch. 2.6 - Prob. 17E Ch. 2.6 - Prob. 18E Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Use the method discussed under Bernoulli Equations...Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Prob. 26E Ch. 2.6 - Prob. 27E Ch. 2.6 - Prob. 28E Ch. 2.6 - Use the method discussed under Equations with...Ch. 2.6 - Prob. 30E Ch. 2.6 - Prob. 31E Ch. 2.6 - Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - Prob. 35E Ch. 2.6 - In Problems 3340, solve the equation given in: 36....Ch. 2.6 - Prob. 37E Ch. 2.6 - Prob. 38E Ch. 2.6 - Prob. 39E Ch. 2.6 - Prob. 40E Ch. 2.6 - Prob. 41E Ch. 2.6 - Prob. 42E Ch. 2.6 - Prob. 43E Ch. 2.6 - Show that equation (13) reduces to an equation of...Ch. 2.6 - Prob. 45E Ch. 2.6 - Prob. 46E Ch. 2.6 - Prob. 47E Ch. 2.6 - Prob. 48E Ch. 2 - In Problems 130, solve the equation. 1....Ch. 2 - Prob. 2RP Ch. 2 - Prob. 3RP Ch. 2 - Prob. 4RP Ch. 2 - Prob. 5RP Ch. 2 - In Problems 130, solve the equation. 6. 2xy3 dx ...Ch. 2 - In Problems 130, solve the equation. 7. t3y2 dt +...Ch. 2 - Prob. 8RP Ch. 2 - In Problems 130, solve the equation. 9. (x2 + y2)...Ch. 2 - Prob. 10RP Ch. 2 - Prob. 11RP Ch. 2 - Prob. 12RP Ch. 2 - Prob. 13RP Ch. 2 - Prob. 14RP Ch. 2 - Prob. 15RP Ch. 2 - Prob. 16RP Ch. 2 - Prob. 17RP Ch. 2 - Prob. 18RP Ch. 2 - Prob. 19RP Ch. 2 - Prob. 20RP Ch. 2 - Prob. 21RP Ch. 2 - Prob. 22RP Ch. 2 - Prob. 23RP Ch. 2 - In Problems 130, solve the equation. 24. (y/x +...Ch. 2 - Prob. 25RP Ch. 2 - Prob. 26RP Ch. 2 - Prob. 27RP Ch. 2 - Prob. 28RP Ch. 2 - Prob. 29RP Ch. 2 - Prob. 30RP Ch. 2 - Prob. 31RP Ch. 2 - Prob. 32RP Ch. 2 - Prob. 33RP Ch. 2 - Prob. 34RP Ch. 2 - Prob. 35RP Ch. 2 - Prob. 36RP Ch. 2 - Prob. 37RP Ch. 2 - Prob. 38RP Ch. 2 - Prob. 39RP Ch. 2 - Prob. 40RP Ch. 2 - Prob. 41RP

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