An apple farm yields an average of 33 bushels of apples per tree when 19 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?

To find the optimal number of trees to plant per acre for maximum yield, we can set up a function based on the information provided. Let \( x \) be the number of additional trees planted beyond the initial 19 trees. Therefore, the total number of trees planted per acre is \( 19 + x \). The yield per tree decreases by 1 bushel for each additional tree planted. Thus, the yield per tree when \( x \) additional trees are planted is: \[ 33 - x \] The total yield \( Y \) (in bushels) can be expressed as: \[ Y = \text{(number of trees)} \times \text{(yield per tree)} = (19 + x)(33 - x) \] Expanding this expression: \[ Y = (19 + x)(33 - x) = 19 \cdot 33 - 19x + 33x - x^2 \] \[ Y = 627 + 14x - x^2 \] This is a quadratic function in the standard form \( Y = -x^2 + 14x + 627 \). The graph of this function is a downward-opening parabola, and the maximum yield occurs at the vertex. The \( x \)-coordinate of the vertex of a quadratic function \( ax^2 + bx + c \) is given by: \[ x = -\frac{b}{2a} \] In our case, \( a = -1 \) and \( b = 14 \): \[ x = -\frac{14}{2 \cdot -1} = \frac{14}{2} = 7 \] Thus, the optimal number of additional trees to plant is \( x = 7 \). Therefore, the total number of trees to plant per acre is: \[ 19 + x = 19 + 7 = 26 \] So, the optimal number of trees to plant on an acre to achieve the highest yield is **26 trees**.