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

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?

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.

You want to find Pr[-2 < Z < -1].

The table tells you that

• Pr[Z < 0] = 0.5000

• Pr[Z < 1.00] = 0.8412

• Pr[Z < 2.00] = 0.9772

• Pr[Z < 3.00] = 0.9987

We have

Pr[-2 < Z < -1] = Pr[Z < -1] - Pr[Z < -2]

(because the distribution of Z is continuous)

… = Pr[Z > 1] - Pr[Z > 2]

(by symmetry of the distribution about its mean)

… = (1 - Pr[Z < 1]) - (1 - Pr[Z < 2])

(by definition of complement)

… = Pr[Z < 2] - Pr[Z < 1]

… = 0.9772 - 0.8412

… = 0.1360 ≈ 0.14 … … … (B)