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Math Advanced Math Fundamentals of Differential Equations (9th Edition)

(a) To find: The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F . The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F will be 82.2 minutes _ . Given: Initially, the temperature of the blood plasma is 40 ° F . The plasma can be used only at 90 ° F . The oven temperature is 120 ° F and it takes 45 min for the plasma to warm to 90 ° F . Calculation: Let T be the temperature of the blood plasma and M be the temperature of the oven. 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) is applicable 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 ) Since the temperature of the blood plasma is 40 ° F at t = 0 and the oven temperature is 120 ° F , substitute T = 40 , M = 120 , and t = 0 in the above solution to find the constant C . T = M + C e − k t 40 = 120 + C e − k ( 0 ) 40 − 120 = C C = − 80 Substitute C = − 80 in equation (1) and simplify further. T = 120 − 80 e − k t ... ( 2 ) Since the temperature of the blood plasma is 90 ° F at t = 45 minutes and the oven temperature is 120 ° F , substitute t = 45 and T = 90 in the above solution to find the constant k . 90 = 120 − 80 e − k ( 45 ) − 30 = − 80 e − 45 k − 30 − 80 = e − 45 k 3 8 = e − 45 k Take logarithm on both sides. ln ( 8 3 ) = 45 k ln ( 8 3 ) = 45 k k = ln ( 8 3 ) 45 k = 0.0218 Substitute k = 0.0218 in equation (1) and simplify further. T ( t ) = M + C e − 0.0218 t ... ( 3 ) This is a general equation for the temperature of the blood plasma. Now, find the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F . Since the temperature of blood plasma is 40 ° F at t = 0 and the oven temperature is 100 ° F , substitute T = 40 , M = 100 , and t = 0 in the equation T = M + C e − 0.0218 t to find the constant C . 40 = 100 + C e − 0.0218 ( 0 ) − 60 = C C = − 60 Substitute C = − 60 in the equation T = 100 + C e − 0.0218 t and simplify further. T = 100 − 60 e − 0.0218 t To calculate the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F , substitute T = 90 in the above equation and simplify further. T = 100 − 60 e − 0.0218 t 90 = 100 − 60 e − 0.0218 t − 10 = − 60 e − 0.0218 t − 10 − 60 = e − 0.0218 t Simplify further as follows. 1 6 = e − 0.0218 t 6 = e 0.0218 t ln | 6 | = 0.0128 t t = ln 6 0.0128 t ≈ 82.2 Thus, the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F will be 82.2 minutes _ . (b) To find: The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F . The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F will be 31.8 minutes _ . Since the temperature of blood plasma is 40 ° F at t = 0 and the oven temperature is 140 ° F , substitute T = 40 , M = 140 , and t = 0 in the equation T = M + C e − 0.0218 t to find the constant C . 40 = 140 + C e − 0.0218 ( 0 ) 40 − 140 = C e − 0.0218 ( 0 ) − 100 = C Substitute C = − 100 in the equation T = 140 + C e − 0.0218 t and simplify further. T = 140 − 100 e − 0.0218 t To calculate the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F , substitute T = 90 in the above equation and simplify further. T = 140 − 100 e − 0.0218 t 90 = 140 − 100 e − 0.0218 t 90 − 140 = − 100 e − 0.0218 t − 50 = − 100 e − 0.0218 t − 50 − 100 = e − 0.0218 t Simplify further as follows. 1 2 = e − 0.0218 t 2 = e 0.0218 t ln | 2 | = 0.0128 t t ≈ 31.8 Thus, the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F will be 31.8 minutes _ . (c) To find: The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 80 ° F . The expected temperature is never attained. Since the temperature of blood plasma is 40 ° F at t = 0 and the oven temperature is 80 ° F , substitute T = 40 , M = 80 , and t = 0 in the equation T = M + C e − 0.0218 t to find the constant C . 40 = 80 + C e − 0.0218 ( 0 ) 40 − 80 = C e − 0.0218 ( 0 ) − 40 = C Substitute C = − 40 in the equation T = 80 + C e − 0.0218 t and simplify further. T = 80 − 40 e − 0.0218 t To calculate the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 80 ° F , substitute T = 80 in the above equation and simplify further. 90 = 80 − 40 e − 0.0218 t 10 = − 40 e − 0.0218 t 1 e − 0.0218 t = − 40 10 e 0.0218 t = − 4 It is not possible that the exponential can be negative. It means that the expected temperature is never attained.

(a) To find: The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F . The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F will be 82.2 minutes _ . Given: Initially, the temperature of the blood plasma is 40 ° F . The plasma can be used only at 90 ° F . The oven temperature is 120 ° F and it takes 45 min for the plasma to warm to 90 ° F . Calculation: Let T be the temperature of the blood plasma and M be the temperature of the oven. 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) is applicable 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 ) Since the temperature of the blood plasma is 40 ° F at t = 0 and the oven temperature is 120 ° F , substitute T = 40 , M = 120 , and t = 0 in the above solution to find the constant C . T = M + C e − k t 40 = 120 + C e − k ( 0 ) 40 − 120 = C C = − 80 Substitute C = − 80 in equation (1) and simplify further. T = 120 − 80 e − k t ... ( 2 ) Since the temperature of the blood plasma is 90 ° F at t = 45 minutes and the oven temperature is 120 ° F , substitute t = 45 and T = 90 in the above solution to find the constant k . 90 = 120 − 80 e − k ( 45 ) − 30 = − 80 e − 45 k − 30 − 80 = e − 45 k 3 8 = e − 45 k Take logarithm on both sides. ln ( 8 3 ) = 45 k ln ( 8 3 ) = 45 k k = ln ( 8 3 ) 45 k = 0.0218 Substitute k = 0.0218 in equation (1) and simplify further. T ( t ) = M + C e − 0.0218 t ... ( 3 ) This is a general equation for the temperature of the blood plasma. Now, find the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F . Since the temperature of blood plasma is 40 ° F at t = 0 and the oven temperature is 100 ° F , substitute T = 40 , M = 100 , and t = 0 in the equation T = M + C e − 0.0218 t to find the constant C . 40 = 100 + C e − 0.0218 ( 0 ) − 60 = C C = − 60 Substitute C = − 60 in the equation T = 100 + C e − 0.0218 t and simplify further. T = 100 − 60 e − 0.0218 t To calculate the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F , substitute T = 90 in the above equation and simplify further. T = 100 − 60 e − 0.0218 t 90 = 100 − 60 e − 0.0218 t − 10 = − 60 e − 0.0218 t − 10 − 60 = e − 0.0218 t Simplify further as follows. 1 6 = e − 0.0218 t 6 = e 0.0218 t ln | 6 | = 0.0128 t t = ln 6 0.0128 t ≈ 82.2 Thus, the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 100 ° F will be 82.2 minutes _ . (b) To find: The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F . The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F will be 31.8 minutes _ . Since the temperature of blood plasma is 40 ° F at t = 0 and the oven temperature is 140 ° F , substitute T = 40 , M = 140 , and t = 0 in the equation T = M + C e − 0.0218 t to find the constant C . 40 = 140 + C e − 0.0218 ( 0 ) 40 − 140 = C e − 0.0218 ( 0 ) − 100 = C Substitute C = − 100 in the equation T = 140 + C e − 0.0218 t and simplify further. T = 140 − 100 e − 0.0218 t To calculate the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F , substitute T = 90 in the above equation and simplify further. T = 140 − 100 e − 0.0218 t 90 = 140 − 100 e − 0.0218 t 90 − 140 = − 100 e − 0.0218 t − 50 = − 100 e − 0.0218 t − 50 − 100 = e − 0.0218 t Simplify further as follows. 1 2 = e − 0.0218 t 2 = e 0.0218 t ln | 2 | = 0.0128 t t ≈ 31.8 Thus, the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 140 ° F will be 31.8 minutes _ . (c) To find: The time taken for the plasma to warm to 90 ° F if the oven temperature is set at 80 ° F . The expected temperature is never attained. Since the temperature of blood plasma is 40 ° F at t = 0 and the oven temperature is 80 ° F , substitute T = 40 , M = 80 , and t = 0 in the equation T = M + C e − 0.0218 t to find the constant C . 40 = 80 + C e − 0.0218 ( 0 ) 40 − 80 = C e − 0.0218 ( 0 ) − 40 = C Substitute C = − 40 in the equation T = 80 + C e − 0.0218 t and simplify further. T = 80 − 40 e − 0.0218 t To calculate the time taken for the plasma to warm to 90 ° F if the oven temperature is set at 80 ° F , substitute T = 80 in the above equation and simplify further. 90 = 80 − 40 e − 0.0218 t 10 = − 40 e − 0.0218 t 1 e − 0.0218 t = − 40 10 e 0.0218 t = − 4 It is not possible that the exponential can be negative. It means that the expected temperature is never attained.

Solution Summary: The author calculates the time taken for the plasma to warm to 90°F if the oven temperature is set at 100°.

Fundamentals of Differential Equations (9th Edition)

Fundamentals of Differential Equations (9th Edition)

9th Edition

ISBN: 9780321977069

Author: R. Kent Nagle, Edward B. Saff, Arthur David Snider

Publisher: PEARSON

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Fundamentals of Differential Equations (9th Edition)

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. 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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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