Question

The annual yield per walnut tree is fairly constant at 140 pounds per tree when the number of trees per acre is 20 or fewer. For each additional tree over 20​, the annual yield per tree for all trees on the acre decreases by 2 pounds due to overcrowding. How many walnut trees should be planted per acre to maximize the annual yield for the​ acre? What is the maximum number of pounds of walnuts per​ acre?

Answer 1

Answer:

a. To maximize the annual yield for the​ acre the number of walnut to be planted per acre is n = 45

b. the maximum number of pounds of walnuts per​ acre is T = 4,050

Step-by-step explanation:

Let T represent the total number of pounds of walnuts per​ acre and n represent the number of trees per acre.

From the question;

For each additional tree over 20​, the annual yield per tree for all trees on the acre decreases by 2 pounds due to overcrowding

The number of walnut yield per tree can be represented as;

x = 140 - (n-20)2

the total number of pounds of walnuts per​ acre T is;

T = n(x)

T = n(140 - (n-20)2) = 140n - 2n(n-20) = 140n - 2n^2 + 40n

T = 180n - 2n^2

Maximizing T, we will differentiate T.

T is maximum at dT/dn = 0

dT/dn = 180 - 4n = 0

4n = 180

n = 180/4 = 45

Substituting the value of into the equation of T;

T = 180(45) - 2(45^2)

T = 4,050

Therefore,

a. To maximize the annual yield for the​ acre the number of walnut to be planted per acre is n = 45

b. the maximum number of pounds of walnuts per​ acre is T = 4,050