an apple farm yields an average of 37 bushels of apples per tree when 25 trees are planted on an acre of…

an apple farm yields an average of 37 bushels of apples per tree when 25 trees are planted on an acre of ground. each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. how many trees should be planted on an acre in order to get the highest yield? in order to get the highest yield, trees should be planted on an acre.

an apple farm yields an average of 37 bushels of apples per tree when 25 trees are planted on an acre of ground. each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. how many trees should be planted on an acre in order to get the highest yield? in order to get the highest yield, trees should be planted on an acre.

Answer

Explanation:

Step1: Define variables

Let (x) be the number of additional trees planted. The number of trees per acre is (n = 25 + x). The yield per tree is (y=37 - x).

Step2: Formulate the yield function

The total yield (Y) (bushels per acre) is the product of the number of trees and the yield per tree. So (Y=(25 + x)(37 - x)). Expand the function: (Y = 25\times37-25x+37x - x^{2}=925 + 12x - x^{2}), or (Y=-x^{2}+12x + 925).

Step3: Find the vertex of the quadratic function

For a quadratic function (Y = ax^{2}+bx + c) ((a=-1), (b = 12), (c = 925)), the (x) - coordinate of the vertex is given by (x=-\frac{b}{2a}). Substitute (a=-1) and (b = 12) into the formula: (x=-\frac{12}{2\times(-1)} = 6).

Step4: Calculate the number of trees

Since (n=25 + x), substitute (x = 6) into the equation. So (n=25+6=31).

Answer:

(31)