An apple orchard has 30 trees per acre and average yield is 400 apples per tree .For each additional tree planted per acre , yield per tree reduces by 10 apples . How many trees per acre will give the largest crop of apples ? What is the yield of apples per acre ?

Dear Student,

Please find below the solution to the asked query :

Let the number of new trees planted per acre = xNow, number of trees in 1 acre = 30 + xNow, each new tree will reduce the average yield per tree by 10 apples. So each tree will now give 400-10x apples.So, we have 30+x trees per acre each of which gives 400-10x apples.When x new trees are planted per acre, then there will be a total of 30+x400-10x apples.Let y = 30+x400-10xy = 12000-300x+400x-10x2y = 12000+100x-10x2dydx = 100-20xFor maxima or minima, we get100-20x = 0x = 5Now, d2ydx2 = -20 < 0So, y is maximum at x = 5Hence, yield is maximum , when 5 new trees are planted per acre.Now, yield of apples per acre = yat x = 5 = 30+5400-10×5 = 35×350 = 12250

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