Question

An apple orchard has an average yield of 32 bushels of apple per acre. For each unit increase in tree density, yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield

Answer 1

The optimal number of trees per acre in an apple orchard to maximize yield is not given by the linear function provided, since the yield decreases as tree density increases. The maximum yield would occur at the lowest possible tree density that is still viable for the orchard. However, practical orchard management considerations such as pollination and fruiting requirements would influence the actual optimal density.

Step-by-step explanation:

The student is asking for the optimal number of trees per acre to plant in an apple orchard to maximize yield. Given that the orchard has an average yield of 32 bushels of apple per acre and the yield decreases by 2 bushels for each additional tree planted, we can set up a function to represent the total yield based on the tree density (number of trees).

Let X be the number of trees per acre. The yield per tree will decrease by 2 bushels for each additional tree, so the yield per acre can be modeled as Y = 32 - 2X. To find the maximum yield, we need to find the value of X that maximizes Y.

However, since we have a linear function and not a quadratic one, this function will continuously decrease as X increases, indicating that the maximum yield per acre is at the minimum tree density. Therefore, to maximize the yield, we would technically not increase the tree density at all beyond the minimum necessary for a viable orchard. This seems counterintuitive, as orchards require a certain number of trees to produce fruits, so there is likely a practical lower limit to the density that is not accounted for in this simplified equation. Practical considerations, such as the minimum number of trees needed for successful pollination and fruiting, are beyond the scope of this math problem but would need to be considered to find a truly optimal number of trees per acre.

Answer 2

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 26 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize the yield?

Answer:

Step-by-step explanation:

From the given information:

Let assume that 26+x trees per acre are planted

then the yield per acre will be (26+x)(32-2x)

However;

As x = 0 (i.e. planting 26 per acre), we have;

= (26+0) (32 - 2 (0))

= 26 × 32

= 832

As x = 1 (i.e planting 19 per acre), we have:

= (26+1) (32-2(1)

= 27 × 30

= 810

As x = 2 (i.e. planting 20 per acre), we have:

= (26 +2 ) ( 32 - 2(2)

= 28 × 28

= 784

The series continues in a downward direction for the yield per acre.

Thus, for maximum plant 19 per acre, it can achieved by method of calculus given that the differentiation of the maximum point of x = 1

Finally, due to integer solution, it is not advisable to use calculus as such other methods should be applied.