Question

A farmer finds that if she plants 60 trees per acre, each tree will yield 45 bushels of fruit. she estimates that for each additional tree planted per acre, the yield of each tree will decrease by 4 bushels. how many trees should she plant per acre to maximize her harvest?

Answer 1

To maximize the harvest, the farmer should plant 60 trees per acre.

Step-by-step explanation:

Let's assume x is the number of additional trees planted per acre.

The total number of trees planted per acre when x additional trees are planted is 60 + x.

The yield per tree when x additional trees are planted is given by the equation 45 - 4x.

To maximize the harvest, we need to find the value of x that maximizes the yield per acre.

The yield per acre is calculated by multiplying the number of trees planted per acre with the yield per tree:

Yield per acre = (60 + x) * (45 - 4x)

To find the maximum value of the yield per acre, we take the derivative of the equation and set it equal to zero:

(60 + x)(-4) + (45 - 4x)(1) = 0

-240 - 4x + 45 - 4x = 0

-8x - 195 = 0

-8x = 195

x ≈ -24.375

Since the number of trees planted cannot be negative, we round x down to the nearest whole number:

x = -24

Therefore, the farmer should plant 60 trees per acre to maximize her harvest.

Answer 2

Answer: 60 trees

Explanation:
Even if there are trees planted per acre, the decrease of yield of each tree is significant enough to make it less worthwhile to have more trees than less.

In other words, when she plants 60 trees per acre, she gets 2700 bushels of fruit per acre, because 45 bushels per tree = 45 • 60 = 2700. If you plant another tree, you get the equation 41 • 61 = 2501 because there are 4 less bushels per tree. 2501 < 2700 so 60 per acre would maximize the harvest.