revenue

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August 28th 2008, 09:50 AM
boondocksaint202

revenue

Please Help I am completely confused on how to do this. Thanks

REVENUE : An apple orchard produces annual revenue of $60 per tree when planted with 100 trees. Because of overcrowding, the annual revenue per tree is reduced by $0.50 for each additional tree planted.
Assuming http://webwork.math.ttu.edu/webwork2...0ddd72c201.png, write an equation for the revenue produced by http://webwork.math.ttu.edu/webwork2...dd0b8b8e91.png trees:
http://webwork.math.ttu.edu/webwork2...17760d0aa1.png
How many trees should be planted to maximize the revenue from the orchard?
Now suppose that the cost of maintaining each tree is $5 per year.
Write the profit function in terms of http://webwork.math.ttu.edu/webwork2...dd0b8b8e91.png:
http://webwork.math.ttu.edu/webwork2...e31d24ae81.png
How many trees should be planted to maximize the profit from the orchard?
August 28th 2008, 10:32 AM
moolimanj

This is an example of an optimisation problem using linear programming. Have a look at this and see if it makes sense:

[html]http://en.wikipedia.org/wiki/Linear_programming[/html]
August 28th 2008, 01:42 PM
Soroban

Hello, boondocksaint202!

Quote:

An apple orchard produces annual revenue of $60 per tree when planted with 100 trees.
Because of overcrowding, the annual revenue per tree is reduced by $0.50
for each additional tree planted.

(a) Assuming $x \geq 100$ , write an equation for the revenue produced by $x$ trees.

For every tree over 100, the revenue per tree is reduced by $\$\frac{1}{2}$

The overage is $x - 100$ .
This reduced the revenue to: . $60 - \frac{1}{2}(x-100)$ dollars per tree.

The Revenue is: .[no. of trees ] x [revenue per tree]

. . $R \;=\;x\bigg[60 - \frac{1}{2}(x - 100)\bigg] \quad\Rightarrow\quad \boxed{R \;=\;110x - \frac{1}{2}x^2}$

Quote:

(b) How many trees should be planted to maximize the revenue from the orchard?

The Revenue function is: . $R \;=\;110x - \frac{1}{2}x^2$

This is a down-opening parabola; its maximum is at its vertex.
. . The vertex is at: . $x \:=\:\frac{-b}{2a}$

We have: . $a = \text{-}\frac{1}{2},\;b = 110$

Hence: . $x \:=\:\frac{\text{-}110}{2\left(\text{-}\frac{1}{2}\right)} \:=\:110$

Therefore, 110 trees should be planted for maximum revenue.

Quote:

(c) Suppose that the cost of maintaining each tree is $5 per year.
. . Write the profit function in terms of $x$

For $x$ trees, the cost is: . $5x$ dollars.

. . $\text{Profit} \;=\; \text{Revenue} - \text{Cost}$

. . . $P \;=\;\left(110x - \frac{1}{2}x^2\right) - 5x \quad\Rightarrow\quad\boxed{ P\;=\;105x - \frac{1}{2}x^2}$

Quote:

(d) How many trees should be planted to maximize the profit from the orchard?

The profit function is: . $P\;=\;105x - \frac{1}{2}x^2$

This is a down-opening parabola ... with its maximum at its vertex.

The vertex is at: . $x \:=\:\frac{\text{-}105}{2\left(\text{-}\frac{1}{2}\right)} \;=\;105$

Therefore, 105 trees should be planted for maximum profit.