Final answer:
To maximize the apple harvest, the apple grower needs to plant 45 trees per acre. This is determined by finding the vertex of the quadratic function that represents the total yield given the number of trees planted and their corresponding yield.
Step-by-step explanation:
The question is looking to find the optimal number of apple trees to plant per acre to maximize harvest. This is an optimization problem that can be solved using quadratic functions.
Let's define the number of additional trees planted as x. The total number of trees per acre would be 40 + x. If each additional tree reduces the yield by 5 bushels, then the yield per tree would be 250 - 5x bushels. Therefore, the total yield Y per acre can be given by the function:
Y = (40 + x)(250 - 5x) = 10000 + 250x - 5x² - 200x
Simplifying this we get:
Y = -5x² + 50x + 10000
To maximize the yield, we need to find the vertex of this parabola. The x-coordinate of the vertex is given by -b/(2a), where a is the coefficient of x² and b is the coefficient of x in the quadratic function.
Thus, x = -50 / (2 * -5) = -50 / -10 = 5
The grower should plant 40 + 5 = 45 trees per acre to maximize her harvest.