When 30 orange trees are planted per acre each tree yields 150 oranges For each additional
tree per acre, the yield decreases by 3 oranges per tree. Express the total yield of oranges per
acre, Y , as a function of the number of trees planted per acre, x, if x �< or equal to 30.
Let x be the number of additional trees planted per acre
Number of trees per acre = 30+x
yield per tree = 150 - 3x
total = (30+x)(150-3x)
= 4500 + 60x - 3x^2
d(total)/dx = 60 -6x
=0 for a max total
6x=60
x = 10
So the max yield is obtained with 40 trees per acre.
BUT, your condition says that the number of trees per acre is ≤ 30
So I guess 30 would be the max
(strange question!)
the answer they give is y=240x -3x^2 and i have no idea how they got that because i did get the y=4500 +60x-3x^2 as my answer and that is a choice but its not the correct one.
Looks like you defined x as the number of extra trees to be planted, which is what I did as well
They defined x as the actual number of trees planted.
When you differentiate their equation , set it equal to zero and solve you get x = 40
the total number of trees is 40
Notice that is exactly the same answer that I got.
But then the question said that the x ≤ 30, so the max is 30
As I said before, this is a dumb question.
x is suppose to be the number of trees planted per acre,
oh sorry i see now. thanks!
Ok, let's do it their way, (and get the same answer as I got before)
Let the number of actual trees planted be k
(I will now switch my original definitions to this new definition)
number of trees = k , I had x+30
so k= x+30 and x = k-30
yield per acre:
I had 150-3x
= 150 - 3(k-30)
= 150- 3k +90
= 240 - 3k
total yield = k(240-3k)
= 240k - 3k^2
heh, I simply used k instead of their x
so yield = 240x - 3x^2
(remember my use of x is not the same as their use of x )
This will produce a maximum value of x = 40
for a maximum number of trees as 40 per acre, the same as before in my first reply.
But, again, they said there was a max of 30, which is what the original was.
So again, dumb question.
To express the total yield of oranges per acre, Y, as a function of the number of trees planted per acre, x, we need to consider two factors:
1. The initial yield of 150 oranges per tree when 30 orange trees are planted per acre.
2. The decrease in yield of 3 oranges per tree for each additional tree planted per acre beyond the initial 30 trees.
Let's break down the problem step by step:
Step 1: Determine the initial yield of oranges when 30 trees are planted.
When 30 trees are planted per acre, each tree yields 150 oranges. Therefore, the initial yield when x = 30 is:
Initial yield = 30 trees * 150 oranges/tree = 4500 oranges
Step 2: Determine the decrease in yield for each additional tree planted.
For each additional tree per acre, the yield decreases by 3 oranges per tree. Since the initial yield was based on 30 trees, we need to consider the number of additional trees beyond 30 (x - 30) and calculate the decrease in yield:
Decrease in yield = 3 oranges/tree * (x - 30) trees
Step 3: Express the total yield of oranges per acre as a function of the number of trees planted.
To express the total yield of oranges per acre, Y, we add the initial yield to the decrease in yield:
Y = Initial yield - Decrease in yield
Substituting the values we obtained earlier:
Y = 4500 oranges - 3 oranges/tree * (x - 30) trees
This is the expression for the total yield of oranges per acre, Y, as a function of the number of trees planted per acre, x, for x ≤ 30.