Final answer:
To obtain the maximum yield per acre, the cherry grower should plant 30 trees per acre, as it is the number that optimizes the trade-off between the number of trees and the yield per tree reduction, resulting in a total of 960 pounds per acre.
Step-by-step explanation:
The student is asking how to determine the optimal number of cherry trees per acre for maximum yield, which is a Mathematics problem involving optimization. We can start by defining variables for the number of trees per acre (let's use T) and the reduced yield per tree for each additional tree planted (Y). Initially, T=25 and Y=37 pounds. Each additional tree per acre reduces the yield per tree by 1 pound. The relationship between the total yield per acre (Yield) and the number of trees (T) can be represented by the equation:
Yield = T × (37 - (T - 25))
The objective is to maximize the Yield. If we plug T=25, 22, 30, and 20 into the equation, the only meaningful yields will be obtained with T values between 25 and 47 because if you plant more than 47 trees the yield per tree becomes zero or negative, which is not possible in practical scenarios.
To find the optimal number of trees, we need to maximize the equation, which is a quadratic expression and can be analyzed by completing the square or using calculus. The maximum point occurs when T=31 (25 plus half of the 12-pound range reduction from 37 to 25), meaning 31 trees per acre will give the maximum yield. If you plant more than 31 trees per acre, both the yield per tree and the total yield will start decreasing. Thus the correct answer is:
c. Plant 30 trees per acre for the maximum yield.
The maximum yield can be found by plugging back T=30 into the equation yielding:
Yield = 30 × (37 - (30 - 25)) = 30 × 32 = 960 pounds per acre.
This yields the maximum potential harvest, hence letter c is the final answer.