Question

The number of apples produced by each tree in an apple orchard depends on how densely the rees are planted. If n trees are planted on an acre of land, then each tree produces 900−9n apples. So, the number of apples produced per acre is A(n)=900n−9n² Using calculus, find how many trees should be planted in order to obtain the maximum yield of apples. What is the maximum yield?

Answer 1

To find the number of trees that should be planted in order to obtain the maximum yield of apples, we take the derivative of the function A(n)=900n−9n² and set it equal to zero. Solving for n, we find that 50 trees should be planted. The maximum yield of apples is 22,500.

Step-by-step explanation:

To find the number of trees that should be planted in order to obtain the maximum yield of apples, we need to find the critical points of the function A(n)=900n−9n². To do this, we can take the derivative of the function and set it equal to zero. Let's proceed with the steps:

1. Find the derivative of A(n) with respect to n: A'(n) = 900 - 18n.
2. Set A'(n) equal to zero and solve for n: 900 - 18n = 0. Simplifying, we get 18n = 900, and then n = 50.
3. This critical point represents a potential maximum. We can verify this by checking the second derivative of A(n). If A''(n) is negative at n = 50, then the critical point is a maximum.
4. Find the second derivative of A(n): A''(n) = -18.
5. A''(n) is negative for all values of n. Therefore, the critical point at n = 50 is indeed a maximum.

So, in order to obtain the maximum yield of apples, 50 trees should be planted. To find the maximum yield, we substitute n = 50 into the function A(n): A(50) = 900(50) - 9(50)² = 45,000 - 22,500 = 22,500 apples.