Download Lesson Plan #6

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Lesson Plan #27
Date: Wednesday November 14th, 2011
Class: AP Calculus
Topic: Optimization Problems
Aim: How do we solve optimization problems?
Objectives:
1) Students will be able to solve problems where they have to maximize or minimize a value.
HW# 27:
1) The sum of one number and two times a second number is 24. What numbers should be selected so that their
product is as large as possible?
2) The product of two positive numbers is 192. What numbers should be chosen so that the sum of the first plus three
times the second is a minimum?
Do Now:
You run a small tutoring school. The graph at
right represents the amount of profit you take in
per week depending on the number of students
you have. Based on the graph, approximately
how many students should you take in to
maximize your profit?
With respect to the topics we have discussed, what
do we call the location where this maximum
occurs?
In general, what do we do to find this relative
maximum?
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Example:
Find two positive numbers that minimize the sum of twice the first number plus the second if the product of the two numbers is
288.
Solution:
𝑥 = 1𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑦 = 2𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑆 = 2𝑥 + 𝑦, 𝑥𝑦 = 288
Solve for 𝑥  𝑥
=
288
𝑦
Substitute in 𝑆 = 2𝑥 + 𝑦  𝑆 = 2 (
′
Differentiate  𝑆 = −
576
𝑦2
288
𝑦
)+𝑦 𝑆 =
576
𝑦
+𝑦
+1
Set equal to zero to find critical numbers  −
576
𝑦2
+ 1 = 0  𝑦 = 24, 𝑥 = 12
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Example 1:
The difference between two numbers is 50. Find the two numbers so that their product is as small as possible
Example 2:
Find two positive numbers whose sum is 110 and whose product is a maximum
Example 3:
The sum of one positive number and twice a second positive number is 100. Find the two numbers so that their product is a
maximum
Example 4:
A rectangle has a perimeter of 100 feet. What length and width should it have so that its area is maximum?
Example 5:
Find the dimensions of a cone (height and radius) of maximum volume if the diameter and height of the cone total 24 cm.
Example 6:
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The graph of 𝑦 = − 𝑥 + 2 encloses a region with the x-axis and y-axis in the first quadrant. A rectangle in the enclosed region
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1
has a vertex at the origin and the opposite vertex on the graph of f 𝑦 = − 𝑥 + 2. Find the dimensions of the rectangle so that its
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area is maximum.
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Sample Test Questions:
1) A line is drawn through the point (1,2) forming a right triangle with the positive x- and y-axes. The slope
of the line forming the triangle of least area is
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A) -1
B) -2
C) -4
D) 
E) -3
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2) The area of the largest isosceles triangle that can be drawn with one vertex at the origin and with the
others on a line parallel to and above the x-axis and on the curve y  27  x 2 is
A) 12 3
B) 27
C) 24 3
D) 54
E) 108
Free Response:
3) If the perimeter of an isosceles triangle is 18cm , find the maximum area of the triangle.
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If enough time:
4) A net enclosure for golf practice is open at one end, as shown in the figure. Find the dimensions that require the
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least amount of netting if the volume of the enclosure its to be 83 cubic meters.
x
x
y
5) An indoor physical fitness room consists of a rectangular region with a semicircle at each end. If the perimeter of
the room is to be a 200-meter running track, find the dimensions, to the nearest tenth of a meter, of the rectangle
that will make the area of the rectangular region as large as possible.
6) Find the dimensions of the rectangle with the greatest area that can be inscribed in the semicircle
y  9  x2