An orchard owner is buying 4.09 acres of land to plant more trees. he figures he will plant 10 trees per acre. how many trees will he plant on his new land?

Step-by-step explanation:

 Find the function that describes the per-tree yield, Y, in terms of x.  

A) Y = 1100 ... if x is no more than 45 trees per acre  

B) Y = 1100 - 30*(x-45) = 2450 - 30x if x is greater than 45 trees per acre  

Find the total yield per acre, T, that results from planting x trees per acre.  

C) T =1100*x ...if x is no more than 45 trees per acre  

D) T = (2450 - 30x)*x = 2450x - 30x² if x is greater than 45 trees per acre  

Differentiate T with respect to x  

E) dT / dx =1100 ...if x is less than 45 trees per acre  

F) dT / dx = 2450 - 60x ...if x is greater than 45 trees per acre  

G) Does this derivative ever equal zero?  

The first part 1100 is never equal to zero while  

2450 - 60x = 0 ==> x = 2450/60 = 40.833 < 45  

so dT/dx is never equal to zero.  

Moreover when x>45, 2450 - 60x <0 .  

So the function T is increasing for x <45 and decreasing  

for x>45. The maximum can be at x = 45.  

H) Optimal value of x :45 trees per acre  

I) Maximum yield : 1100*45 = 49500 peaches per acre  

By the way, the same yield is 2450*45 - 30*45² = 49500.  

J) Is T differentiable when x equals 45?  

T is not differentiable at x = 45, because  

(x--> 45 - ) lim dT/dx = 1100  

(x--> 45 + ) lim dT/dx = 2450 - 60*45 = -250  

1100=/= -250

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