Max Total Yield: Plant 37-38 Apple Trees

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An orchard owner has determined that planting 25 apple trees yields an average of 500 apples per tree, but each additional tree reduces the yield by 10 apples per tree. To maximize total yield, calculations show that the optimal number of trees to plant is between 37 and 38. The quadratic equation derived from the yield calculations indicates that planting 12.5 additional trees beyond the initial 25 maximizes yield, rounding to 38 trees total. Planting more than this number will lead to decreased overall yield. Therefore, for maximum total yield, 37 or 38 apple trees should be planted.
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.. an orchard owner has maintained records that show that if 25 appl trees are planted in one acre then each tree yields an aveerage of 500 apples. The yield decreases by 10 apples per tree for each additional tree that is planted. How many trees should be planted for max total yield?

The answer is 37 or 38. The answer I get is 12.5...

trees planted * yield of each tree = total apples
(25+x)(500-10x)
12500 - 250x + 500x - 10x^2
-10x^2 + 250x + 12500

-b/2a = 12.5

I don't understand how the answer would be less than the minimum (25) trees that were planted initially
edit: sorry, stupid mistake, 25+12.5 = 37.5 ... forgot what i was solving for
 
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Your calculation of 12.5 trees is incorrect. The question is asking for the maximum total yield, not just the maximum number of trees. To find the maximum total yield, we need to find the number of trees that will give us the maximum number of apples. This can be done by finding the vertex of the parabola created by the equation -10x^2 + 250x + 12500.

Using the formula -b/2a, we get -250/(-20) = 12.5. This means that the maximum yield will occur when 12.5 trees are planted. However, since we cannot plant half a tree, we round up to the nearest whole number, giving us 13 trees.

Therefore, to achieve the maximum total yield, the orchard owner should plant 25+13 or 38 apple trees. Planting any more trees will result in a decrease in total yield.
 

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