An apple farm yields an average of 40 bushels of apples per tree when 26 trees are planted on an acre of ground. Each time 1 more tree is planted per​ acre, the yield decreases by 1 bushel​ (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest​ yield?

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Part 1

In order to get the highest​ yield, 
  
enter your response here trees should be planted on an acre.

Let \( x \) be the number of additional trees planted beyond the initial 26. The total number of trees planted would then be \( 26 + x \). 

The yield per tree decreases by 1 bushel for each additional tree, so the yield per tree becomes \( 40 - x \).

The total yield, \( Y(x) \), is the product of the number of trees and the yield per tree:

\[
Y(x) = (26 + x)(40 - x)
\]

Now, expand this expression:

\[
Y(x) = 26(40 - x) + x(40 - x) = 1040 - 26x + 40x - x^2
\]

Simplifying:

\[
Y(x) = 1040 + 14x - x^2
\]

To maximize the total yield, we need to find the value of \( x \) that maximizes \( Y(x) \). This can be done by taking the derivative of \( Y(x) \) with respect to \( x \) and setting it equal to 0:

\[
\frac{dY}{dx} = 14 - 2x
\]

Set the derivative equal to 0 to find the critical point:

\[
14 - 2x = 0
\]

\[
x = 7
\]

So, the optimal number of additional trees is \( x = 7 \). Since the total number of trees planted is \( 26 + x \), the optimal number of trees to plant is:

\[
26 + 7 = 33
\]

Thus, **33 trees** should be planted to get the highest yield.

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the yield affected if 5 additional trees are planted instead of 7?
2. What happens to the total yield if more than 33 trees are planted?
3. How would the yield change if the initial yield per tree was 50 bushels instead of 40?
4. How do we verify that 33 trees give a maximum yield and not a minimum?
5. What role does the derivative play in finding the optimal yield?

**Tip:** When optimizing problems, look for how variables change with respect to each other and use derivatives to identify maximum or minimum points.

An apple farm yields an average of 40 bushels of apples per tree when 26 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?

Learn how to determine the optimal number of trees to plant on an acre of land to maximize apple yield. This problem uses quadratic functions and derivative calculus to find the best solution, ideal for high school students studying algebra and optimization techniques.

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