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?

I mostly need help getting an equation and defining a variable

the number of apples is yield/tree * # trees.

With x trees, yield per tree is 200 - 5(x-50) for x > 50

So, total crop is

c(x) = x(200-5(x-50)) = x(450-5x) = 450x - 5x^2
c'(x) = 450-10x
c' = 0 at x=45

So, the max yield is achieved with 45 trees

Well, in this case, let's define the variable "x" as the number of additional trees planted in the orchard. The initial number of trees is 50, so the total number of trees in the orchard would be 50 + x.

Now, the average number of apples produced per tree drops by 5 for each additional tree planted. So the average number of apples per tree would be 200 - 5x.

To find the total number of apples produced in the orchard, we can multiply the total number of trees by the average number of apples per tree:

Total apples = (50 + x) * (200 - 5x)

Now, we can simplify and expand this equation:

Total apples = 10,000 + 200x - 250x - 5x^2

Simplifying this expression, we get:

Total apples = -5x^2 - 50x + 10,000

To find the optimal number of trees to plant in the orchard, we need to maximize the total number of apples. One way to do this is by finding the value of "x" that gives us the highest point on the parabolic graph of the equation above.

Equation is derived, now let's hope the apples don't get too "core-ny" with the math!

now if you are allowed to remove trees, and for each tree removed, the average goes up by 5, then the optimal is to remove five trees.

Dang - forgot the restriction on the domain.

To find the optimal number of trees to plant in the orchard, let's define a variable. Let's say "x" represents the number of additional trees planted in the orchard.

Now, let's consider how the number of apples produced is affected by the additional trees. Each tree, on average, produces 200 apples each year. However, for each additional tree planted, the average number of apples drops by 5.

So, for the first 50 trees, the average number of apples produced is 200. For the next tree, the average number of apples would be 200 - 5 = 195. For the second additional tree, it would be 195 - 5 = 190, and so on.

Therefore, the average number of apples produced, with "x" additional trees, can be expressed as: 200 - 5x.

To determine the optimal number of trees, we need to maximize the total number of apples produced. This can be calculated by multiplying the average number of apples per tree with the total number of trees.

The total number of trees can be represented as: 50 (original trees) + x (additional trees).

So, the equation for the total apples produced, with "x" additional trees, would be: (200 - 5x) * (50 + x).

To find the optimal number of trees to plant, we need to find the value of "x" that maximizes this equation. We can do this by taking the derivative of the equation and finding where it equals zero. However, since this is a programming environment, I would recommend using a numerical method or a graphing calculator to solve for "x" and find the optimal number of trees to plant in the orchard.

number apples=average*number trees

let x be the nubmer of trees 50<x<inf

number apples=(200-(x-50)*5)(x) where x is the number of trees, 50<x<infinity
N=200x-5x^2 +250x
dN/dx=0=200-10x+250
10x=450
x=45

but x>50, so look at optimal
check x=50 N=50*200=10000 apples
check x=51 N=(51)(195)=9945

so indeed, the optimal is 50.