An apple orchard yields 20 bushels of apples per tree when there are 75 trees per acre. For each additional tree per acre, it is estimated that yield will decrease 3 bushels per tree. How many trees should be planted per acre to maximize the yield? Round to the nearest tree.
If there are x additional trees, then the yield is 20-3x per tree
So the total yield is
(75+x)(20-3x)
Now just find the vertex of that parabola as usual.
Hmmm. I suspect a typo, since after only 7 additional trees, the yield per tree will be negative.
present yield: 20 bushels per tree
present number of trees: 75 per acre
income = 20(75) = 1500 bushels
Let the number of additional trees planted be n
yield = 20 - 3n, (20-3x > 0) ----> n < 20/3
number of trees = 75+n
income = (20-3n)(75 + n)
= 1500 - 205n - 3n^2
using Calculus:
d(income) = -205 - 6n
= 0 mor a max income
6n = 205
n = appr 34.2
But that makes no physical sense, the yield would be negative number of bushels
looking at the graph of income = (20-3n)(75+n) shows that the current situation gives
you the maximum yield, unless you take away trees
https://www.wolframalpha.com/input/?i=plot+y+%3D+%3D+%2820-3x%29%2875+%2B+x%29
eg
n trees yield income
1 75 20 1500
2 76 17 1292
3 77 14 1078
-1 74 23 1702
To maximize the yield of the apple orchard, we need to find the number of trees per acre that will give us the highest total yield.
Let's assume that x is the number of additional trees per acre.
For each additional tree per acre, the yield will decrease by 3 bushels per tree. Therefore, the yield per tree for x additional trees will be 20 - 3x.
The total number of trees per acre will be 75 + x.
The total yield per acre will be the product of the number of trees per acre and the yield per tree:
Total Yield = (75 + x)(20 - 3x)
To find the maximum yield, we need to find the value of x that maximizes the total yield.
One way to find the maximum value of the total yield is by graphing the equation (75 + x)(20 - 3x) and finding the x-coordinate of the vertex. However, since we need to round to the nearest tree, it would be more appropriate to use a numerical method.
We can use a method called "differentiation" to find the x-coordinate of the point where the total yield is maximized.
Differentiating the equation (75 + x)(20 - 3x) with respect to x, we get:
d(Total Yield)/dx = 20 - 6x - 3(75 + x)
Simplifying this equation, we get:
d(Total Yield)/dx = 20 - 6x - 225 - 3x
d(Total Yield)/dx = -6x - 3x - 205
To find the critical points where the derivative equals zero, we set -6x - 3x - 205 = 0:
-9x - 205 = 0
-9x = 205
x = -205/9
Since we are looking for the number of trees, we round x to the nearest whole number:
x = -205/9 ≈ -22.8 ≈ -23 (rounded to the nearest whole number)
Since it doesn't make sense to have a negative number of additional trees, we discard this solution.
Therefore, we conclude that the number of trees to maximize yield is 0 additional trees per acre, resulting in a total of 75 trees per acre.
To find the number of trees that should be planted per acre to maximize the yield, we need to determine the point at which the yield starts decreasing due to increasing the number of trees.
Let's assume the number of additional trees per acre is represented by the variable x. Therefore, we can express the total number of trees per acre as 75 + x.
As stated in the problem, the estimated decrease in yield per additional tree is 3 bushels per tree.
So, the decrease in yield due to the additional trees would be 3x bushels.
To maximize the yield, we need to find the point at which the decrease in yield equals the initial yield of 20 bushels per tree. This can be represented by the equation:
20 - 3x = 0
To solve for x, we can rearrange the equation to isolate x:
3x = 20
x = 20/3
x ≈ 6.67
Since we are looking for the number of trees, x should be rounded to the nearest whole number.
Thus, approximately 6.67 additional trees per acre are required to reach the maximum yield. It is not possible to plant a fraction of a tree, so we can round this value to the nearest whole number.
Therefore, we should plant 7 additional trees per acre to maximize the yield.