A grower estimates that if he picks his apple crop now, he will obtain 1000 boxe

Question

A grower estimates that if he picks his apple crop now, he will obtain 1000 boxes of apples, which he can sell at $3 per box. However, he thinks his crop will increase by 120 boxes of apples for each week he delays picking, but that the price will drop at a rate of 15 cents per box per week; in addition, he estimates that approximately 20 boxes per week will spoil for each week he delays picking. When should he pick his crop to obtain the largest total cash return? How much will he receive for his crop at that time?

Explanation / Answer

Cash return = [1000 + (120-20)n](3-0.15n) 1000 is the thousand boxes (120 - 20) is the extra 120 boxes per week delay minus the 20 that will spoil per week (3 -0.15n) is the price each box will sell for. Helpful problem solving hint: check your equation with a few easy values Check equation: if he doesn't wait n = 0 and the equation becomes Cash return = 3000 - yes this is correct if he waits two weeks, there will be 240 extra boxes minus 40 spoiled boxes and each will sell for $2.70 1,200(2.70) = $3,240 [1000 + (120-20)2](3 - 0.15{2}) = 3240, so the equation looks good. Now, how do we want to maximize this? non calculus way: put it into spreadsheet weeks cash 0 3000 1 3135 2 3240 3 3315 4 3360 5 3375 6 3360 7 3315 8 3240 9 3135 10 3000 11 2835 12 2640 13 2415 14 2160 15 1875 16 1560 17 1215 18 840 19 435 20 0 Five weeks is the best and $3,375 is the cash return Calculus way multiply out equation 3,000 -150n +300n -15 n^2 3,000 +150n - 15n^2 d cash/dn = 150 - 30n set derivative equal to zero 30n= 150 n = 150/30 = 5 substitute 5 into equation [1,000 + 500](3-.15{5}) = $3,375 So, five weeks and $3,375 using calculus. Helpful problme solving hint: try to solve the problem in different ways.

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