Explanation Given: The amount of money collected when pears are picked per tree is 30 c and each tree will give average of 60 pounds of pears per week. Also, if there is an increase in the quantity of pears by 6 pounds, then the average price is reduced by 2 c . Procedure used: 1. Select the variables and find the some relations among the variables. 2. Form a mathematical model as shown below. Maximize ( or minimize ) f ( x ) on the interval I . 3. Obtain the critical numbers of f ( x ) . 4. Use second derivative test to check for absolute maximum (or minimum) and find the absolute maximum (or minimum) of f ( x ) on the interval I and the numbers x as the absolute maximum must occur either at the critical points or at the endpoints with the help of substitution. 5. Find the values of the left variables with the help of the solution obtained in step 4. Calculation: Suppose the number of 2 c price reductions be x . Thus, the quantity of pears per tree becomes 60 + 6 ⋅ x = 60 + 6 x . The average price per pound becomes 30 − 2 ⋅ x 100 = 0.3 − 0.02 x . Obtain the yield function as follows. Yield=Quantity of pears per tree × average price per pound Y ( x ) = ( 60 + 6 x ) × ( 0.3 − 0.02 x ) Y ( x ) = 18 − 1.2 x + 1.8 x − 0.12 x 2 Y ( x ) = 18 + 0.6 x − 0.12 x 2 As the price is always nonnegative, the endpoint becomes, 0.30 − 0.02 x ≥ 0 − 0.02 x ≥ − 0.30 2 x ≤ 30 x ≤ 15 In order to find the maximum yield, the problem that should be computed as, Maximize Y ( x ) = 18 + 0

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Chapter 4.6, Problem 30E

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To find: The time when the pear trees should be picked so that it maximizes the return and the maximum return.

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Chapter 4.6, Problem 29E

Chapter 4.6, Problem 31E

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The time when the pear trees should be picked so that it maximizes the return and the maximum return.

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