Explanation Given Information: The sunspot activity can be modeled by the function N ( t ) = 57.7 sin [ 0.602 ( t − 1.43 ) ] +58 .8 where t is the number of years since January 1 , 1997 and N ( t ) is the number of sunspots observed at time t. Formula Used: The total change of the function from x = a to x = b is, ∫ a b f ( x ) d x . Where, function f ( x ) is the rate of change of quantity. The average of a function f ( x ) on an interval [ a , b ] is, f ¯ = 1 b − a ∫ a b f ( x ) d x . The sum and difference rules for an integral is, ∫ [ f ( x ) ± g ( x ) ] d x = ∫ f ( x ) d x ± ∫ g ( x ) d x Where, f ( x ) and g ( x ) are two functions. The formula for the integration of the function is, ∫ sin ( a x + b ) d x = − 1 a x + b cos ( a x + b ) + C , where the function is f ( x ) = sin ( a x + b ) , and a and b are constants, with a ≠ 0 . Constant multiple rule is ∫ k ⋅ g ( x ) d x = k ∫ g ( x ) d x where k is a constant. Calculation: Consider the provided function, N ( t ) = 57.7 sin [ 0.602 ( t − 1.43 ) ] +58 .8 . Here, N ( t ) is the number of sunspots observed since January 1 , 1997 where t = 0 and for January 1 , 2002 , t = 5 , Determine the total number of sunspots visible over the 2 year period from 5 to 7 using definite integral, N ( t ) = ∫ 5 7 ( 57.7 sin [ 0.602 ( t − 1.43 ) ] +58 .8 ) d t Then determine the average of a function N ( t ) on an interval [ a , b ] by applying the formula f ¯ = 1 b − a ∫ a b f ( x ) d x . N ¯ = 1 7 − 5 ∫ 5 7 57.7 sin [ 0.602 ( t − 1.43 ) ] +58 .8 d t N ¯ = 1 2 ∫ 5 7 57.7 sin [ 0.602 ( t − 1.43 ) ] +58 .8 d t To evaluate the integral 1 2 ∫ 5 7 57.7 sin [ 0.602 ( t − 1.43 ) ] +58 .8 d t , suppose that the function f ( u ) is ∫ 5 7 57.7 sin [ 0.602 ( t − 1.43 ) ] d t and g ( u ) is ∫ 5 7 58.8 d t . Apply the sum rule, ∫ [ f ( u ) + g ( u ) ] d u = ∫ f ( u ) d u + ∫ g ( u ) d u 1 2 ∫ 5 7 57

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Chapter 16.3, Problem 63E

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To calculate: The average number of sunspots visible over the 2-year period beginning January 1, 2002 where the activity of the sun fluctuates in cycles of around 10−11 years.

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Chapter 16.3, Problem 62E

Chapter 16.3, Problem 64E

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Chapter 16 Solutions

EBK FINITE MATH AND APPLIED CALCULUS

Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E

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The average number of sunspots visible over the 2 -year period beginning January 1 , 2002 where the activity of the sun fluctuates in cycles of around 10 − 11 years.

Solution Summary: The author calculates the average number of sunspots visible over the 2 year period beginning January 1, where the activity of the sun fluctuates in cycles of around 10-11 years.

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To calculate: The average number of sunspots visible over the 2-year period beginning January 1, 2002 where the activity of the sun fluctuates in cycles of around 10−11 years.

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Chapter 16 Solutions