# revenue

• Aug 28th 2008, 09:50 AM
boondocksaint202
revenue

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? • Aug 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] • Aug 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 $0.50 for each additional tree planted. (a) Assuming$\displaystyle x \geq 100$, write an equation for the revenue produced by$\displaystyle x$trees. For every tree over 100, the revenue per tree is reduced by$\displaystyle \$\frac{1}{2}$

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

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

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

Quote:

(b) How many trees should be planted to maximize the revenue from the orchard?
The Revenue function is: .$\displaystyle R \;=\;110x - \frac{1}{2}x^2$

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

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

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

Therefore, 110 trees should be planted for maximum revenue.

Quote:

(c) Suppose that the cost of maintaining each tree is $5 per year. . . Write the profit function in terms of$\displaystyle x$For$\displaystyle x$trees, the cost is: .$\displaystyle 5x$dollars. . .$\displaystyle \text{Profit} \;=\; \text{Revenue} - \text{Cost}$. . .$\displaystyle P \;=\;\left(110x - \frac{1}{2}x^2\right) - 5x \quad\Rightarrow\quad\boxed{ P\;=\;105x - \frac{1}{2}x^2}$Quote: (d) How many trees should be planted to maximize the profit from the orchard? The profit function is: .$\displaystyle P\;=\;105x - \frac{1}{2}x^2$This is a down-opening parabola ... with its maximum at its vertex. The vertex is at: .$\displaystyle x \:=\:\frac{\text{-}105}{2\left(\text{-}\frac{1}{2}\right)} \;=\;105\$

Therefore, 105 trees should be planted for maximum profit.