The owner of an orange grove estimates that 150 orange trees per acre will each yield on average 800 oranges per year. For each additional tree planted per acre the number of oranges produced by each tree decreases by 10 per year.
How many trees should be planted per acre in order to obtain the most oranges?
To determine the number of trees that should be planted per acre to obtain the most oranges, we need to find the optimal number of trees where the total number of oranges produced is maximized.
Let's perform a step-by-step analysis:
Step 1: Calculate the total number of trees per acre for each scenario.
- Starting with 150 trees per acre, let's assume the number of additional trees to be planted is represented by 'x'.
- Therefore, the total number of trees per acre would be: 150 + x.
Step 2: Calculate the average number of oranges produced per tree for each scenario.
- The average number of oranges produced per tree decreases by 10 for each additional tree planted.
- So, for each scenario, the average number of oranges produced per tree would be: 800 - (10x).
Step 3: Calculate the total number of oranges produced per acre for each scenario.
- To obtain the total number of oranges produced per acre, we multiply the total number of trees per acre by the average number of oranges produced per tree.
- Therefore, the total number of oranges produced per acre for each scenario would be: (150 + x) * (800 - 10x).
Step 4: Find the maximum value.
- To find the maximum number of oranges produced per acre, we need to determine the value of 'x' that maximizes the equation (150 + x) * (800 - 10x).
- We can achieve this by taking the derivative of the equation with respect to 'x' and setting it equal to zero.
- Let's differentiate the equation:
d[(150 + x) * (800 - 10x)]/dx = 0
(800 - 10x) + (150 + x) * (-10) = 0
Simplifying the equation:
800 - 10x - 1500 - 10x = 0
-20x = -700
x = -700 / -20
x = 35
Step 5: Calculate the number of trees per acre for maximum oranges.
- Now that we have the value of 'x' as 35, we can calculate the optimal number of trees per acre by adding this to the initial number of trees:
Number of trees per acre = 150 + x
= 150 + 35
= 185
Therefore, to obtain the most oranges per acre, you should plant 185 trees per acre.
To determine the number of trees that should be planted per acre in order to obtain the most oranges, we need to consider the trade-off between the number of trees and the number of oranges produced per tree.
Step 1: Find the total number of oranges produced with 150 trees per acre.
The owner estimates that each tree will yield an average of 800 oranges per year. Therefore, with 150 trees per acre, the total number of oranges produced would be:
Total number of oranges = Number of trees * Oranges per tree
Total number of oranges = 150 trees * 800 oranges per tree
Total number of oranges = 120,000 oranges
Step 2: Calculate the decrease in oranges per year for each additional tree.
For each additional tree planted per acre, the number of oranges produced by each tree decreases by 10 per year. Therefore, the decrease in oranges per additional tree would be 10.
Step 3: Calculate the number of trees that would result in the same total number of oranges.
Let's assume that x represents the number of additional trees planted per acre. To find the point where the number of additional oranges produced from the extra trees cancels out the decrease from each tree, we can set up the following equation:
Total number of oranges = 120,000 oranges
Number of trees * (Oranges per tree - Decrease in oranges per additional tree) = Total number of oranges
(150 + x) * (800 - 10x) = 120,000
Step 4: Solve the equation to find the value of x.
Multiply out the equation:
(150 + x)(800 - 10x) = 120,000
120,000 + 150x - 1,500x - 10x^2 = 120,000
Combine like terms:
-10x^2 - 1,350x + 120,000 = 0
Divide the equation by -10 to simplify:
x^2 + 135x - 12,000 = 0
Step 5: Solve the quadratic equation.
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring is not straightforward, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 1, b = 135, and c = -12,000. Plugging these values into the quadratic formula, we get:
x = (-135 ± √(135^2 - 4(1)(-12,000))) / (2(1))
x = (-135 ± √(18,225 + 48,000)) / 2
x = (-135 ± √(66,225)) / 2
x = (-135 ± 257.23) / 2
Using the positive value for x:
x = (-135 + 257.23) / 2
x = 122.23 / 2
x ≈ 61.1
Step 6: Calculate the total number of trees.
To find the total number of trees that should be planted per acre to obtain the most oranges, we add the original number of trees (150) to the additional number of trees (x):
Total number of trees = 150 + x
Total number of trees = 150 + 61.1
Total number of trees ≈ 211.1
Therefore, the owner should plant approximately 211 trees per acre in order to obtain the most oranges.