Mini-Project 1 (1)
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School
Houston Community College *
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Course
3336
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
8
Uploaded by ColonelFreedom8712
M 408Q Miniature Project 1 - Hungry for Apples?
Name and UTEID:
————————————————————
Name and UTEID:
————————————————————
Name and UTEID:
————————————————————
The main difference between this assignment and the discussion worksheets is that your
submission for this assignment is expected to be as polished as possible: the document you
submit should be typed and organized; your MATLAB plots should include titles and axis
labels, with all required features clear and distinct.
Having recently purchased one acre of pristine land in Bloomington, Indiana, on Earth
C-137, you are going into business for yourself: with
Jerry Smith
as your chief marketing
executive, you intend to plant an orchard full of apple trees in order to break into the apple
game. More apples sold means more money for ice creams, so you want to maximize your
apple crop to maximize profit.
In other words, you need to determine the answer to the
following question: if an apple orchard occupies one acre of land, how many trees should
it contain so as to produce the largest apple crop? This is an example of an
optimization
problem
. In this project, you will solve the problem using a simple mathematical model of
the relationship between the number of apple trees planted and the apple yield per tree.
It might seem natural to simply plant as many apple trees as your one acre will accommodate,
following the reasoning that more trees means more apples, but there is a flaw in this logic:
if there are too many trees in a single acre, they crowd together and compete for resources.
Each tree then gets less sunlight and fewer nutrients, and, as a result, produces fewer apples.
For example, the relationship between the number of trees,
N
, and the yield per tree,
Y
(
N
)
,
may be reflected by the following graph:
1
Problem
1
.
In words, describe what the graph in the figure above says about apple yield per
tree.
2
Problem
2
.
Our goal is to choose
N
so that the total apple yield,
T
(
N
)
, will be as large as
possible. Give an equation relating
T
(
N
)
, Y
(
N
)
, and
N
.
Since the graph of
Y
(
N
)
given in the figure on the first page isn’t described by a simple
algebraic formula, let’s replace it with the following approximation (close is good enough
amirite?):
For the remaining problems, use the figure above (not the one on the first page).
3
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Problem
3
.
Derive a formula for the portion of the graph of
Y
(
N
)
above the interval
[40
,
180]
consisting of all
N
with
40
≤
N
≤
180
. Show all of your work.
4
Problem
4
.
Give a formula for
T
(
N
)
on the interval
[40
,
180]
.
5
Problem
5
.
Fill in the following blanks:
T
(
N
) =
if
0
≤
N
≤
40
if
40
< N <
180
if
N
≥
180
.
6
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Problem
6
.
Use MATLAB to produce a plot of the graph of
T
(
N
)
on
[0
,
180]
(you need to
include an image of your plot for this problem). Your plot should include a title and labels
on each axis.
7
Problem
7
.
Using your graph for
T
(
N
)
, estimate the maximum possible total apple yield.
How many apple trees should be planted to achieve the maximum yield?
8