Modelling the orchard yields with diminishing returns per treeSylvia has an apple orchard. One season, her 100 trees yielded 140 apples per tree. She wants to increase her production by adding more trees to the orchard. However, she knows that for every 10 additional trees she plants, she will lose 4 apples per tree (i.e., the yield per tree will decrease by 4 apples). How many trees should she have in the orchard to maximize her production of apples?

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zakos2zn1mr · Accepted Answer

Let&amp;nbsp;t=the number of treesLet&amp;nbsp;y=the yield per treeWe know that when&amp;nbsp;t=100,y=140. Furthermore, we know that for every 10 additional trees, the yield decreases by 4. I&#039;m going to assume that this means that for every additional tree, the yield decreases by&amp;nbsp;0.4&amp;nbsp;(obviously this is an average yield). If I increase the number of trees by M, then t and y change as follows:t=100+My=140−0.4MWe want the total yield to be as large as possible. This means that ty has to be as large as possible:ty=(100+M)(140−0.4M)&amp;nbsp;=14000−40M+140M−0.4M2&amp;nbsp;=−0.4M2+100M+14000&amp;nbsp;=−0.4(M2−250M−35000)Completing the square on&amp;nbsp;M2−250M−35000, we haveM2−250M−35000=(M−125)2−1252−35000&amp;nbsp;=(M−125)2−50,625Hence,&amp;nbsp;ty=−0.4[(M−125)2−50,625]&amp;nbsp;=−0.4(M−125)2+20,250Since&amp;nbsp;−0.4(M−125)2&amp;nbsp;is always non-positive, the maximum yield is 20,250, which occurs when&amp;nbsp;M=125. Thus, the total number of trees in the orchard should be 225.

Modelling the orchard yields with diminishing returns per

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