Explanation Given: The size A ( t ) of a radioactive sample is measured at various time intervals. time t (in hours) 0 1 3 4 6 number of bacteria A ( t ) (in grams) 100 24.5 1.5 0.4 0.1 Table ( 1 ) Approach: Let the data points be ( x 1 , y 1 ) ; ( x 2 , y 2 ) ; ... ; ( x m , y m ) . Use the following steps to obtain the equation of the least squares parabola: The required equation is of the form P ( t ) = C e k t where C and k are constants. Take the natural logarithm both sides of above equation. ln P ( t ) = k t + ln C Take y ( t ) = ln P ( t ) , b = ln C so that y ( t ) = k t + b Write A = [ x 1 1 x 2 1 ⋮ ⋮ x m 1 ] and x = [ k b ] . Using least square method, x 0 = [ k 0 b 0 ] and C 0 = e b 0 . The least square solution is given by P ( t ) = C 0 e k 0 t . Calculation: Let the solution for the given data points be A ( t ) = C e k t . The data points are: ( 0 , 100 ) ; ( 1 , 24.5 ) ; ( 3 , 1.5 ) ; ( 4 , 0.4 ) ; ( 6 , 0.1 ) . Then, A = [ 0 1 1 1 3 1 4 1 6 1 ] , x = [ k b ] , y = [ 4.605 3.198 0.405 − 0.916 − 2.302 ] A T A = [ 0 1 3 4 6 1 1 1 1 1 ] [ 0 1 1 1 3 1 4 1 6 1 ] = [ 0 ⋅ 0 + 1 ⋅ 1 + 3 ⋅ 3 + 4 ⋅ 4 + 6 ⋅ 6 0 ⋅ 1 + 1 ⋅ 1 + 3 ⋅ 1 + 4 ⋅ 1 + 6 ⋅ 1 1 ⋅ 0 + 1 ⋅ 1 + 1 ⋅ 3 + 1 ⋅ 4 + 1 ⋅ 6 1 ⋅ 1 + 1 ⋅ 1 + 1 ⋅ 1 + 1 ⋅ 1 + 1 ⋅ 1 ] = [ 62 14 14 5 ]

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 5.4, Problem 16P

To determine

The estimation for A(t) at any time t using least square method for the given date.

time t (in hours) 0 1 3 4 6

number of bacteria A(t) (in grams) 100 24.5 1.5 0.4 0.1

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Chapter 5.4, Problem 15P

Chapter 5.4, Problem 17P

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Chapter 5 Solutions

Differential Equations and Linear Algebra (4th Edition)

Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - True-False Review For Questions a-g, decide if the...Ch. 5.1 - Prob. 6TFR Ch. 5.1 - Prob. 7TFR Ch. 5.1 - Use the standard inner product in 5 to determine...Ch. 5.1 - Prob. 2P Ch. 5.1 - If f(x)=sinx and g(x)=x on [0,], use the function...Ch. 5.1 - If f(x)=sinx and g(x)=2cosx+4 on [0,/2], use the...

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The estimation for A ( t ) at any time t using least square method for the given date. time t (in hours) 0 1 3 4 6 number of bacteria A ( t ) (in grams) 100 24.5 1.5 0.4 0.1

Solution Summary: The author explains how to determine the estimation for A(t) using least square method for the given date.

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The estimation for A(t) at any time t using least square method for the given date.

time t (in hours) 0 1 3 4 6

number of bacteria A(t) (in grams) 100 24.5 1.5 0.4 0.1

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Chapter 5 Solutions

time t (in hours)	0	1	3	4	6
number of bacteria A(t) (in grams)	100	24.5	1.5	0.4	0.1

The estimation for A ( t ) at any time t using least square method for the given date. time t (in hours) 0 1 3 4 6 number of bacteria A ( t ) (in grams) 100 24.5 1.5 0.4 0.1

Solution Summary: The author explains how to determine the estimation for A(t) using least square method for the given date.

Concept explainers

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The estimation for A(t) at any time t using least square method for the given date. time t (in hours) 0 1 3 4 6 number of bacteria A(t) (in grams) 100 24.5 1.5 0.4 0.1

Want to see the full answer?

Chapter 5 Solutions

The estimation for A(t) at any time t using least square method for the given date.

time t (in hours) 0 1 3 4 6

number of bacteria A(t) (in grams) 100 24.5 1.5 0.4 0.1