= Question 4 (5pt): The Orchard Problem. Below is the graph y f(t) of the annual harvest (assumed continuous) in kg/year from my cranapple orchard t years after planting. The trees take about 25 years to get established, and from that point on, for the next 25 years, they give a fairly good yield. But after 50 years, age and disease are taking their toll, and the annual yield is falling off. 40 35 30 。 ៣៩ ថា8 8 8 8 6 25 20 15 10 y 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 The orchard problem is this: when should the orchard be cut down and re- planted, thus starting the cycle again? What you want to do is to maximize your average harvest per year over a full cycle. Of course there are costs to cutting the orchard down and replanting, but it turns out that we can ignore these. The first cost is the time it takes to cut the trees down and replant but we assume that this can effectively be done in a week, and the loss of time is negligible. Secondly there is the cost of the labour to cut and replant, but this is exactly offset by the value of the lumber, which is exceedingly hard with a lovely grain, and prized in cabinet making. (a) Let A(t) be the total yield over the first t years. We work with a continuous time model so that A(T) is the area under the f(t) graph over the interval [0,T]. Find a general expression for the average yield per year over a full cycle T, in terms of A(T). That will be the expression to be maximized. (b) To maximize the expression in (a), set its derivative to zero. Use the funda- mental theorem of calculus to analyze and simplify the general condition for the optimal cycle time T. (c) The above graph of y is given by the formula: f(t) = (t²(60− t))/1000. Given this equation, use the condition found in (b) to find the optimal cycle time T.
= Question 4 (5pt): The Orchard Problem. Below is the graph y f(t) of the annual harvest (assumed continuous) in kg/year from my cranapple orchard t years after planting. The trees take about 25 years to get established, and from that point on, for the next 25 years, they give a fairly good yield. But after 50 years, age and disease are taking their toll, and the annual yield is falling off. 40 35 30 。 ៣៩ ថា8 8 8 8 6 25 20 15 10 y 5 0 0 5 10 15 20 25 30 35 40 45 50 55 60 The orchard problem is this: when should the orchard be cut down and re- planted, thus starting the cycle again? What you want to do is to maximize your average harvest per year over a full cycle. Of course there are costs to cutting the orchard down and replanting, but it turns out that we can ignore these. The first cost is the time it takes to cut the trees down and replant but we assume that this can effectively be done in a week, and the loss of time is negligible. Secondly there is the cost of the labour to cut and replant, but this is exactly offset by the value of the lumber, which is exceedingly hard with a lovely grain, and prized in cabinet making. (a) Let A(t) be the total yield over the first t years. We work with a continuous time model so that A(T) is the area under the f(t) graph over the interval [0,T]. Find a general expression for the average yield per year over a full cycle T, in terms of A(T). That will be the expression to be maximized. (b) To maximize the expression in (a), set its derivative to zero. Use the funda- mental theorem of calculus to analyze and simplify the general condition for the optimal cycle time T. (c) The above graph of y is given by the formula: f(t) = (t²(60− t))/1000. Given this equation, use the condition found in (b) to find the optimal cycle time T.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
Related questions
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Question 4 (5pt): The Orchard Problem. Below is the graph y
f(t) of
the annual harvest (assumed continuous) in kg/year from my cranapple orchard t
years after planting. The trees take about 25 years to get established, and from
that point on, for the next 25 years, they give a fairly good yield. But after 50
years, age and disease are taking their toll, and the annual yield is falling off.
40
35
30
。 ៣៩ ថា8 8 8 8 6
25
20
15
10
y
5
0
0 5 10 15 20 25 30 35 40 45 50 55 60
The orchard problem is this: when should the orchard be cut down and re-
planted, thus starting the cycle again? What you want to do is to maximize your
average harvest per year over a full cycle. Of course there are costs to cutting the
orchard down and replanting, but it turns out that we can ignore these. The first
cost is the time it takes to cut the trees down and replant but we assume that this
can effectively be done in a week, and the loss of time is negligible. Secondly there
is the cost of the labour to cut and replant, but this is exactly offset by the value
of the lumber, which is exceedingly hard with a lovely grain, and prized in cabinet
making.
(a) Let A(t) be the total yield over the first t years. We work with a continuous
time model so that A(T) is the area under the f(t) graph over the interval
[0,T]. Find a general expression for the average yield per year over a full cycle
T, in terms of A(T). That will be the expression to be maximized.
(b) To maximize the expression in (a), set its derivative to zero. Use the funda-
mental theorem of calculus to analyze and simplify the general condition for
the optimal cycle time T.
(c) The above graph of y is given by the formula: f(t) = (t²(60− t))/1000. Given
this equation, use the condition found in (b) to find the optimal cycle time T.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43685a05-72bf-4819-84f1-719cb68e9c1a%2F8a3f29dd-ba70-40a7-afa8-d42d44098e4c%2F3b683_processed.png&w=3840&q=75)
Transcribed Image Text:=
Question 4 (5pt): The Orchard Problem. Below is the graph y
f(t) of
the annual harvest (assumed continuous) in kg/year from my cranapple orchard t
years after planting. The trees take about 25 years to get established, and from
that point on, for the next 25 years, they give a fairly good yield. But after 50
years, age and disease are taking their toll, and the annual yield is falling off.
40
35
30
。 ៣៩ ថា8 8 8 8 6
25
20
15
10
y
5
0
0 5 10 15 20 25 30 35 40 45 50 55 60
The orchard problem is this: when should the orchard be cut down and re-
planted, thus starting the cycle again? What you want to do is to maximize your
average harvest per year over a full cycle. Of course there are costs to cutting the
orchard down and replanting, but it turns out that we can ignore these. The first
cost is the time it takes to cut the trees down and replant but we assume that this
can effectively be done in a week, and the loss of time is negligible. Secondly there
is the cost of the labour to cut and replant, but this is exactly offset by the value
of the lumber, which is exceedingly hard with a lovely grain, and prized in cabinet
making.
(a) Let A(t) be the total yield over the first t years. We work with a continuous
time model so that A(T) is the area under the f(t) graph over the interval
[0,T]. Find a general expression for the average yield per year over a full cycle
T, in terms of A(T). That will be the expression to be maximized.
(b) To maximize the expression in (a), set its derivative to zero. Use the funda-
mental theorem of calculus to analyze and simplify the general condition for
the optimal cycle time T.
(c) The above graph of y is given by the formula: f(t) = (t²(60− t))/1000. Given
this equation, use the condition found in (b) to find the optimal cycle time T.
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