Fruit tree output varies from year to year; can you plan ahead? Part I

Calculus Level 1

You have recently purchased an orchard and it can hold 200 peach trees. Unfortunately, all of the trees are infested with bugs and need to be cut down. You decide to replant, and learn that peach trees have the following characteristics:

for the first 4 years, peach trees produce no fruit.
for years 5-20, peach trees produce 5 bushels / year
after year 20, peach trees no fruit

How often should you plan on replanting new trees to maximize peach production over the next 100 years? (note that fractional years are not an option, since you'd miss the harvest if the trees aren't present for a full year)?

Details and assumptions

If you are interested in optimization problem, you might want to look at Part Two of this problem.

Every 100 years Every 20 years Every 5 years Every 4 years

7 solutions

Chung Gene Keun
Jan 31, 2014

In this question, we want to maximize the total production. This is equivalent to maximizing the average yearly production, if we cut a tree down after $n$ years. Consider the following table:

$\begin{array} { | l | l | l |} \hline \text{Years} & \text{Total output} & \text{Average output} \\ \hline 1 & 0 & 0 \\ 2 & 0 & 0 \\ 3 & 0 & 0 \\ 4 & 0 & 0 \\ 5 & 5 & 1 \\ 6 & 10 & 1.66 \\ 7 & 15 & 2.14 \\ 8 & 20 & 2.5 \\ 9 & 25 & 2.5 \\ 10 & 30 & 3 \\ 11 & 35 & 3.18\\ 12 & 40 & 3.33\\ 13 & 45 & 3.46\\ 14 & 50 & 3.57 \\ 15 & 55 & 3.66\\ 16 & 60 & 3.75\\ 17 & 65 & 3.82\\ 18 & 70 & 3.88\\ 19 & 75 & 3.94\\ 20 & 80 & 4 \\ 21 & 80 & 3.81 \\ n \geq 21 & 80 & \frac{80}{n} \\ \hline \end{array}$

We easily see that the maximum average yearly production occurs when we cut down the tree after 20 years.

you do said that after 20 years there are no fruits at all.. but on the table??

Hafizh Ahsan Permana - 7 years, 4 months ago

The total output table is actually the cumulative frequency table for peach production from beginning and since after 20 years, no fruit is produced, so the total output for the tree remains the same throughout the column, i.e, $\boxed{80}$ and the average output is the table for total average output per year of each tree for $n$ years.

Prasun Biswas - 7 years, 4 months ago

I think what he means is the total output of the tree in its lifetime

Fengyu Seah - 7 years, 4 months ago

it's the total output. So in the table, year 21 and the following years include the output of the previous years.

Lg Asejo - 7 years, 4 months ago

very good solution

Anuj Shikarkhane - 6 years, 11 months ago

Prasun Biswas
Feb 2, 2014

The logic here is very simple. Since the peach trees produce fruit only during years 5-20, so we should replant trees such that the previous plantation yields maximum fruits and this is possible for upto year 20 as after that they produce no fruit. So, it is obvious that we should plant the new peach trees after 20 years so that the previous plantation gives maximum output and thus maximize peach production over the next 100 years. So, the answer is = Every 20 years.

yaa broder..it's logical..:D!

Yash Bhagwat - 7 years, 4 months ago

Joe Bob
Feb 11, 2014

The answer is obvious

Abhishek Virkar
Feb 11, 2014

if we replant the trees after 5 or 4 ,then from observation it is clear that we will not have any profit . & if we replant after 100 years we will have profit only within 5-20 years . if we replant after each 20 years then we will be profit for 75 years out of 100.

Cva Chhakchhuak
Feb 5, 2014

Lucky Guess i think haha

Aashish Patel
Mar 28, 2014

It Is so simple .We don't have to follow Chung Gene Keun.He had a very long and complicated solution.It is common sense that the tree produces no fruits in first 4 years and it produces fuits in the years between 4 to 20 years.And then it does not produce fruits.It is has a common sense to cut down thw trees after 20 years to optimize the the fruit production.

Qamar Raj
Mar 13, 2014

Bcz the trees produce the peech in 20 years and after that tree produce no fruits so the replanting option is 20 years best .

Fruit tree output varies from year to year; can you plan ahead? Part I

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