Question 435322: Researach for a given orchard has shown that, if 100 pear trees are planted, then the annual revenue is $90 per tree. If more trees are planted they have less room to grow and generate fewer pears per tree. As a result, the annual revenue per tree is reduced by $.70 for each additional tree planted. No matter how many trees are planted, the cost of maintaining each tree is $7.40 per year. How many pear trees should be planted to maximize the profit from the orchard tree for one year?
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! : Research for a given orchard has shown that, if 100 pear trees are planted, then the annual revenue is $90 per tree.
If more trees are planted they have less room to grow and generate fewer pears per tree.
As a result, the annual revenue per tree is reduced by $.70 for each additional
tree planted.
No matter how many trees are planted, the cost of maintaining each tree is $7.40 per year.
How many pear trees should be planted to maximize the profit from the orchard tree for one year?
:
Let x = Additional trees planted
:
Profit = Revenue - cost
:
P = [{90-.70x)(100+x)] - 7.50(x+100)
FOIL
f(x) = 9000 + 90x - 70x - .70x^2 - 7.50x - 750
combine like terms
f(x) = 9000 - 750 + 90x - 70x - 7.5x - .70x^2
A quadratic equation
f(x) = -.7x^2 + 12.5x + 8250
Max profit occurs at the axis symmetry; x = -b/(2a)
x =
x =
x = 8.9 ~ 9 additional trees
find the profit when 109 trees are planted, replace x with 9 in the equation
P = -.7(9^2) + 12.5(9) + 8250
P = -56.7 + 112.5 + 8250
P ~ $8,306 profit when you plant 9 additional trees
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