Research for a given orchard has shown that, if 100 pear trees are planted, then the annual revenue is $117 per tree. If more trees are planted, they have less room to grow and gen... Research for a given orchard has shown that, if 100 pear trees are planted, then the annual revenue is $117 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 $0.90 for each additional tree planted. How many trees should be planted to maximize revenue?
Understand the Problem
The question is asking how many pear trees should be planted to maximize annual revenue based on a given formula where the revenue decreases as more trees are planted. We'll need to formulate an equation based on the provided information and then find the optimal number of trees.
Answer
The optimal number of pear trees to plant is 115.
Answer for screen readers
The optimal number of pear trees to plant is 115.
Steps to Solve
- Define the Variables
Let ( x ) be the number of trees planted beyond the initial 100. Thus, the actual number of trees will be ( 100 + x ).
- Revenue Per Tree Function
The revenue per tree decreases by $0.90 for each additional tree. So, the revenue per tree can be expressed as: $$ R(x) = 117 - 0.90x $$
- Total Revenue Function
The total revenue ( T ) is the revenue per tree multiplied by the number of trees: $$ T(x) = (100 + x)(117 - 0.90x) $$
- Expand the Total Revenue Function
Distributing the terms in the total revenue function gives: $$ T(x) = 11700 + 117x - 0.90x^2 - 90x $$ This simplifies to: $$ T(x) = 11700 + 27x - 0.90x^2 $$
- Find the Maximum Revenue
To maximize the total revenue, we take the derivative of the total revenue function and set it to zero: $$ T'(x) = 27 - 1.80x $$ Set the derivative equal to zero: $$ 0 = 27 - 1.80x $$
- Solve for ( x )
Rearranging gives: $$ 1.80x = 27 $$ $$ x = \frac{27}{1.80} = 15 $$
- Calculate Total Number of Trees
Since ( x ) is the number of trees beyond 100: $$ \text{Total Trees} = 100 + x = 100 + 15 = 115 $$
The optimal number of pear trees to plant is 115.
More Information
Planting 115 pear trees maximizes annual revenue based on the decrease in revenue per additional tree planted. The derived total revenue function showed a quadratic form, indicating a maximum point.
Tips
- Forgetting to correctly define the total number of trees as ( 100 + x ).
- Failing to expand the total revenue function correctly can lead to errors in finding the derivative.
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