Thread: 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 , write an equation for the revenue produced by trees:

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 :

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

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]

Hello, boondocksaint202!

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 , write an equation for the revenue produced by trees.

For every tree over 100, the revenue per tree is reduced by

The overage is .
This reduced the revenue to: . dollars per tree.

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

. .

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

The Revenue function is: .

This is a down-opening parabola; its maximum is at its vertex.
. . The vertex is at: .

We have: .

Hence: .


Therefore, 110 trees should be planted for maximum revenue.

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

For trees, the cost is: . dollars.

. .

. . .

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

The profit function is: .

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

The vertex is at: .


Therefore, 105 trees should be planted for maximum profit.