There are 50 apple trees in an orchard, and
each tree produces an average of 200 apples
each year. For each additional tree planted
within the orchard, the average number of
apples produced drops by 5. What is the
optimal number of trees to plant in the
orchard?
To find the optimal number of trees to plant in the orchard, we need to determine the point at which the total number of apples produced is maximized.
Let's break down the problem step-by-step:
1. Calculate the average number of apples produced per tree after each additional tree is planted:
- Starting with 50 trees producing an average of 200 apples each, the total number of apples produced is 50 * 200 = 10,000 apples.
- For each additional tree planted, the average number of apples drops by 5. So, after planting one additional tree, the average becomes 195 apples per tree (200 - 5). After two additional trees, the average becomes 190 apples per tree, and so on.
2. Determine the total number of trees and apples for each scenario:
- With 51 trees, the average number of apples per tree is 195, so the total number of apples is 51 * 195.
- With 52 trees, the average is 190, so the total number of apples is 52 * 190.
- Continue this process for different numbers of trees until we reach a point where adding more trees decreases the total number of apples produced.
3. Compare the total number of apples for each scenario:
- Calculate the total number of apples for each number of trees planted.
- Identify the scenario with the highest total number of apples produced. This will be the optimal number of trees to plant in the orchard.
Let's perform the calculations:
Number of trees: 50
Average apples per tree: 200
Total apples: 50 * 200 = 10,000
Number of trees: 51
Average apples per tree: 195 (200 - 5)
Total apples: 51 * 195 = 9,945
Number of trees: 52
Average apples per tree: 190 (200 - 2 * 5)
Total apples: 52 * 190 = 9,880
By continuing this process, you can compare the total number of apples for each number of trees planted to find the optimal number. In this case, it seems that the optimal number of trees to plant in the orchard is 50, as it yields the highest total number of apples (10,000).