3-7: Optimization Problems

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```3-7: Optimization Problems
Objective:
1. To solve applied
maximum and
minimum problems
Assignment:
• P. 223-227: 1, 9, 11, 13,
18, 19, 21, 23, 24, 27,
44, 56, 57
You will be able to solve
applied maximum and
minimum problems
The Bigger Box
As employees of Lee J.
Monroe, we have been
tasked with constructing a
box with maximum
volume.
The Bigger Box
To make said box, we will
take a sturdy sheet of
cardstock, which
measures 8.5 inches by 11
inches, and cut a square
from each corner, keeping
in mind that the box must
be as voluminous as
possible, and then make a
box from the resulting net.
Optimization Algorithm
Identify all given quantities and quantities
to be determined, and sketch a picture.
Step 1
Write a primary equation for the
quantity to be optimized.
Step 3
Step 2
Reduce the primary equation to one
having a single independent variable
using secondary equations.
Optimization Algorithm
Step 4
Determine a feasible domain for the
primary equation.
Closed Interval Test
Use calculus to determine the
minimum or maximum value.
First Derivative Test
Step 5
Second Derivative Test
Exercise 1
A manufacturer wants
to design an open box
having a square base
and a surface area of
108 square inches.
What dimensions will
produce a box with
maximum volume?
Exercise 2
Which points on the
graph of 𝑦 = 4 − 𝑥 2
are closest to the
point (0,2)?
Exercise 3
A rectangular page is to contain 24 square
inches of print. The margins at the top and
bottom of the page are to be 1 &frac12; inches, and
the margins on the left and right are to be 1
inch. What should the dimensions of the
page be so that the least amount of paper is
used?
Exercise 4
Two posts, one 12 feet high and the other 28
feet high, stand 30 feet apart. They are to be
stayed by two wires, attached to a single
stake, running for ground level to the top of
each post. Where should the stake be
placed to use the least amount of wire?
Exercise 5
Four feet of wire is to be
used to form a square
and a circle. How much
of the wire should be
used for the square and
how much should be
used for the circle to
enclose the maximum
total area?
3-7: Optimization Problems
Objective:
1. To solve applied
maximum and
minimum problems
Assignment:
• P. 223-227: 1, 9, 11, 13,
18, 19, 21, 23, 24, 27,
44, 56, 57
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