Announcements
Dixie Cup
Old Test Problem
Norman Window
More Fun With Optimization
Peter A. Perry
University of Kentucky
November 2, 2016
Row Your Boat
Announcements
Dixie Cup
Old Test Problem
Math of the Day
• Announcements
• Dixie Cup
• Inscribed Rectangle
• Norman Window
• Row, Row, Row Your Boat
Norman Window
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Announcements
Dixie Cup
Old Test Problem
Norman Window
Announcements
1. Your next exam is Tuesday, November 15
2. Tuesday November 8 is a holiday. You are responsible for
completing Worksheet 22. Form study groups now!
3. REEF scoring is not accurate and will be fixed
4. Your sixth and last written assignment is due on Friday,
November 4
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Announcements
Dixie Cup
Old Test Problem
Norman Window
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The Dixie Cup
A Dixie Cup is made by taking a triangular wedge out of a circle of
radius R and gluing the sides of the wedge together. What is the
maximum volume that can be held by the resulting conical cup?
r
R
R
h
Constraint: R 2 = h2 + r 2 , so 0 ≤ h ≤ R
R
Announcements
Dixie Cup
Old Test Problem
Norman Window
The Dixie Cup Problem, Continued
• Constraint: R 2 = h2 + R 2
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Dixie Cup
Old Test Problem
Norman Window
The Dixie Cup Problem, Continued
• Constraint: R 2 = h2 + R 2
• Volume: V =
1 2
πr h
3
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Announcements
Dixie Cup
Old Test Problem
Norman Window
The Dixie Cup Problem, Continued
• Constraint: R 2 = h2 + R 2
1 2
πr h
3
• Use constraint to eliminate r 2 :
• Volume: V =
V =
1
1
π (R 2 − h 2 )h = π R 2 h − h 3
3
3
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Announcements
Dixie Cup
Old Test Problem
Norman Window
The Dixie Cup Problem, Continued
• Constraint: R 2 = h2 + R 2
1 2
πr h
3
• Use constraint to eliminate r 2 :
• Volume: V =
V =
1
1
π (R 2 − h 2 )h = π R 2 h − h 3
3
3
• Maximize V (h ) for 0 ≤ h ≤ R
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Announcements
Dixie Cup
Old Test Problem
Norman Window
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The Dixie Cup Problem, Continued
We’ve computed that
V (h ) =
1
π R 2 − h 2 )h
3
At what value of h does the absolute maximum of this function in
[0, R ] occur?
√
A. h = −R/ 2
√
B. h = R/ 2
√
C. h = R/ 3
D. h = R
Why is your answer correct?
Announcements
Dixie Cup
Old Test Problem
Norman Window
The Dixie Cup Problem, Concluded
1
π R 2 − h2 h
3
√
has an absolute maximum at h = R/ 3.
The maximum volume is
V (h ) =
√
√
1
V (R/ 3) = π R 2 − R 2 /3 R/ 3
3
1 2R 2 R
√
= π
3
3
3
2
= √ πR 3
9 3
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Announcements
Dixie Cup
Old Test Problem
Norman Window
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The Inscribed Rectangle Problem
What is the maximum possible area of a rectangle with base on
the x axis and upper vertices on the graph of y = 9 − x 2 ?
Announcements
Dixie Cup
Old Test Problem
Norman Window
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The Inscribed Rectangle Problem
What is the maximum possible area of a rectangle with base on
the x axis and upper vertices on the graph of y = 9 − x 2 ?
Announcements
Dixie Cup
Old Test Problem
Norman Window
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Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
Announcements
Dixie Cup
Old Test Problem
Norman Window
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Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
Announcements
Dixie Cup
Old Test Problem
Norman Window
Row Your Boat
Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
h
w
Announcements
Dixie Cup
Old Test Problem
Norman Window
Row Your Boat
Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
What is the area of the window?
h
w
Announcements
Dixie Cup
Old Test Problem
Norman Window
Row Your Boat
Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
What is the area of the window?
A = hw + π (w /2)2
h
w
Announcements
Dixie Cup
Old Test Problem
Norman Window
Row Your Boat
Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
What is the area of the window?
A = hw + π (w /2)2
What is the perimeter of
the window?
h
w
Announcements
Dixie Cup
Old Test Problem
Norman Window
Row Your Boat
Norman Window
A Norman window has the shape of a rectangle surmounted by a
semicircle. If the perimeter of the window is 30 ft, find the
dimensions of the window so that the greatest possible amount of
light is admitted.
What is the area of the window?
A = hw + π (w /2)2
What is the perimeter of
the window?
h
P = 2h + w + πw /2
w
Announcements
Dixie Cup
Old Test Problem
Norman Window
Row Your Boat
Row Your Boat
A woman at a point A on the shore of a circular lake with radius 2
mi wants to arrive at point C diametrically oppose A on the other
side of the lake in the shortest possible time. She can walk at the
rate of 4 mi/hr and row a boat at 2 mi/hr. How should she
proceed?
B
A
C