Explanation Given Information: The apple powdery mildew is an epidemic that affects the apple shoots and in a new infection, the percentage of apple shoots infected varies from a low of around 10 % at the start of May to a high of around 60 % 6 months later. Formula Used: The vertical offset which is the height of the baseline is, C = low + high 2 The Amplitude is the height of peak above the baseline, A = High − C . The period or wavelength is the length of each cycle. The angular frequency which is the number of cycles in every interval of length 2 π is, ω = 2 π P . The phase shift is the distance to which the function is shifted horizontally. The general model for the sine function is, f ( x ) = A sin [ ω ( x − α ) ] + C . 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 ) d x = − 1 a cos ( a x ) + C , where the function is f ( x ) = sin ( a x ) , 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 percentage of apple shoots infected varies from a low of around 10 % at the start of May to a high of around 60 % 6 months later. Then determine the vertical offset which is the height of the baseline. C = low + high 2 = 10 + 60 2 = 70 2 = 35 Determine the value of Amplitude which is the height of peak above the baseline. A = high − C = 60 − 35 = 25 Then determine the period or wavelength which is the length of each cycle. P = 12 months Then determine the angular frequency which is the number of cycles in every interval of length 2 π . ω = 2 π P = 2 π 12 = π 6 Then determine the phase shift which is the distance to which the function is shifted horizontally. α = starting month + P 4 Where starting month is May which is 5 start from January. Substitute the values. α = 5+ 12 4 = 5+ 3 = 8 Then substitute the values A = 25 , α = 8 , ω = π 6 and x = t in the sine function, f ( x ) = A sin [ ω ( x − α ) ] + C So, f ( x ) = A sin [ ω ( x − α ) ] + C f ( t ) = 25 sin [ π 6 ( t − 8 ) ] + 35 Therefore, the sine function is f ( t ) = 25 sin [ π 6 ( t − 8 ) ] + 35 . Now, consider the function, F ( t ) = 25 sin [ π 6 ( t − 8 ) ] + 35 . Determine apple shoots affected in the first two months from t = 0 to t = 2 . F ( t ) = ∫ 0 2 25 sin [ π 6 ( t − 8 ) ] + 35 d t Then determine the average of a function F ( t ) on an interval [ a , b ] by applying the formula f ¯ = 1 b − a ∫ a b f ( x ) d x . F ¯ = 1 2 − 0 ∫ 0 2 25 sin [ π 6 ( t − 8 ) ] + 35 d t = 1 2 ∫ 0 2 [ 25 sin [ π 6 ( t − 8 ) ] + 35 ] d t To evaluate the integral 1 2 ∫ 0 2 [ 25 sin [ π 6 ( t − 8 ) ] + 35 ] d t , suppose that the function f ( u ) is ∫ 0 2 25 sin [ π 6 ( t − 8 ) ] d t and g ( u ) is ∫ 0 2 35 d t

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 16.3, Problem 66E

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To calculate: The average percentage to the nearest 0.1% of apple shoots affected in the first 2 months of a year using the sine function of the percentage of apple shoots affected by apple powdery mildew t months since the start of the year.

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Chapter 16.3, Problem 65E

Chapter 16.3, Problem 67E

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Chapter 16 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

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 percentage to the nearest 0.1 % of apple shoots affected in the first 2 months of a year using the sine function of the percentage of apple shoots affected by apple powdery mildew t months since the start of the year.

Solution Summary: The author calculates the average percentage of apple shoots affected by apple powdery mildew in the first 2 months of a year.

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To calculate: The average percentage to the nearest 0.1% of apple shoots affected in the first 2 months of a year using the sine function of the percentage of apple shoots affected by apple powdery mildew t months since the start of the year.

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Chapter 16 Solutions