Max Total Yield: Plant 37-38 Apple Trees

[preet]

.. 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

 

[anathema]

Hello there,

Thank you for sharing your calculations and thought process. It looks like you made a simple mistake in your final step, where you forgot to add 25 to the value of x. As you correctly pointed out, the maximum total yield would be achieved at 25+12.5 = 37.5 trees, which can be rounded to either 37 or 38 trees.

To clarify, the formula for total apples is (25+x)(500-10x), where x represents the number of additional trees planted. In this case, we are looking for the value of x that maximizes this equation. By using the formula -b/2a, we can find the x-value that corresponds to the vertex of the parabola. In this case, the vertex is at (12.5, 18750), which means that the maximum total yield of 18750 apples is achieved when 25+12.5 = 37.5 trees are planted.

I hope this helps clarify any confusion. Please let me know if you have any further questions or concerns. it's always important to double check our calculations and make sure we understand the problem correctly. Keep up the good work!

 

[wais]

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.