Radioactive substances A and B have decay constants of 0.02 and 0.05, respectively. If a mixture of these two substances at time t=0 contains MA grams of A and Mg grams of B, then a model for the total ame of the mixture present at time t is y = Mae -0.02t + Mge -0.05t Suppose the initial amounts Ma and Me are unknown, but a scientist is able to measure the total amount present at several times and records the following points (t, y,) : (10, 21.38), (11, 20.69), (12, 20.02),. (14, 18.82), and (15, 18.34). a. Describe a linear model that can be used to estimate M, and Ma. b. Find the least-squares curve based on the equation y = M,e -0.02t + Mee -0.05t -0.02(10) e -0.05(10) 21.38 e -0.02(11) -0.05(11) e E2 20.69 MA e= E3 Ma X= -0.02(12) -0.05(12) ,y = 20.02 -0.02(14) -0.05(14) 18.82 €4 18.34 e -0.02(15) -0.05(15) e (Type exact answers.) b. The least-squares curve is given by the function y= De -0021 + ( De -0.05t (Round the final answers to two decimal places as needed. Round all intermediate values to four decimal places as needed.)

Radioactive substances A and B have decay constants of 0.02 and 0.05, respectively. If a mixture of these two substances at time t=0 contains MA grams of A and Mg grams of B, then a model for the total ame of the mixture present at time t is y = Mae -0.02t + Mge -0.05t Suppose the initial amounts Ma and Me are unknown, but a scientist is able to measure the total amount present at several times and records the following points (t, y,) : (10, 21.38), (11, 20.69), (12, 20.02),. (14, 18.82), and (15, 18.34). a. Describe a linear model that can be used to estimate M, and Ma. b. Find the least-squares curve based on the equation y = M,e -0.02t + Mee -0.05t -0.02(10) e -0.05(10) 21.38 e -0.02(11) -0.05(11) e E2 20.69 MA e= E3 Ma X= -0.02(12) -0.05(12) ,y = 20.02 -0.02(14) -0.05(14) 18.82 €4 18.34 e -0.02(15) -0.05(15) e (Type exact answers.) b. The least-squares curve is given by the function y= De -0021 + ( De -0.05t (Round the final answers to two decimal places as needed. Round all intermediate values to four decimal places as needed.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Radioactive substances A and B have decay constants of 0.02 and 0.05, respectively. If a mixture of these two substances at time t= 0 contains MA grams of A and MB grams of B, then a model for the total amount y
of the mixture present at time t is y = MAe -0.02t + Mge -0.05t Suppose the initial amounts M, and MR are unknown, but a scientist is able to measure the total amount present at several times and records the
following points (t,, y) : (10, 21.38), (11, 20.69), (12, 20.02), (14, 18.82), and (15, 18.34).
a. Describe a linear model that can be used to estimate MA and Mp.
b. Find the least-squares curve based on the equation y = MAe
- 0.02t
+ Mge
- 0.05t
- 0.02(10)
- 0.05(10)
e
e
21.38
13
- 0.02(11)
- 0.05(11)
73
e = E3
e
e
20.69
M,
B =
MB
X =
- 0.02(12)
- 0.05(12)
20.02
e
e
- 0.02(14)
- 0.05(14)
18.82
e
e
18.34
- 0.02(15)
- 0.05(15)
e
93
(Type exact answers.)
b. The least-squares curve is given by the function y = ( De -0.02t +
(Round the final answers to two decimal places as needed. Round all intermediate values to four decimal places as needed.)
e -0.05t

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