4.7 Applied Optimization
Wed Dec 17
Do Now
Differentiate
1) A(x) = x(20 - x)
2) f(x) = x^3 - 3x^2 + 6x - 12
Optimization
• One of the most useful applications of
derivatives is to find optimal designs
– Most cost efficient, maximized profit, etc
• Finding maximum and minimums solve
these optimzation problems
Optimzation
• 1) Draw a picture (if possible)
• 2) Determine what quantity needs to be
maximized or minimized
• 3) Determine what variables are related to
your max/min
• 4) Write a function that describes the max/min
• 5) Use derivatives to find the max/min
• 6) Solve
Ex 1
• A piece of wire of length L is bent into
the shape of a rectangle. Which
dimensions produce the rectangle of
maximum area?
Ex 2
• Your task is to build a road joining a ranch to
a highway that enables drivers to reach the
city in the shortest time. How should this be
done if the speed limit is 60km/h on the road
and 110km/h on the highway? The
perpendicular distance from the range to the
highway is 30km, and the city is 50km down
the highway
Ex 3
• All units in a 30-unit apartment building are
rented out when the monthly rent is set at r =
$1000/month. A survey reveals that one unit
becomes vacant with each $40 increase in
rent. Suppose each occupied unit costs
$120/month in maintenance. Which rent r
maximizes monthly profit?
More examples if time
Closure
• A three-sided fence is to be built next to a
straight section of river, which forms the 4th
side of the rectangular region. Given 90 ft of
fencing, find the maximum area and the
dimensions of the corresponding enclosure
4.7 Optimization
Thurs Dec 18
• Do Now
• An open box is to be made from a 3 ft by 8 ft piece of
sheet metal by cutting out squares of equal size from
the four corners and bending up the sides. Find the
maximum volume the box can have
HW Review: p.262 #1-59
• n/a
Ex 2
• A three-sided fence is to be built next to a
straight section of river, which forms the 4th
side of the rectangular region. Given 60 ft of
fencing, find the maximum area and the
dimensions of the corresponding enclosure
Ex 3
• A square sheet of cardboard 18 in. on a
side is made into an open box (no top)
by cutting squares of equal size out of
each corner and folding up the sides.
Find the dimensions of the box with the
maximum volume.
You try
• You have 40 (linear) feet of fencing with which to
enclose a rectangular space for a garden. Find the
largest area that can be enclosed with this much
fencing and the dimensions of the corresponding
garden
Example 7.3
• Find the point on the parabola
closest to the point (3, 9)
y =9- x
2
Closure
• Journal Entry: How can we use
derivatives to find an optimal design to a
situation?
• HW: p.262 #1-59 EOO Try to get 6-7
correct (skip any CAS, GU problems)
4.7 Optimization
Fri Dec 19
• Do Now
• A child’s rectangular play yard is to be built
next to the house. To make the three sides of
the playpen, 24 ft of fencing are available.
What should be the dimensions of the sides
to make a maximum area?
HW Review: p.262 #1-59
EOO
Worksheet
• HF
Closure
• Hand in: A box with no top is to be built by
taking a 12 inch by 16 inch sheet of
cardboard and cutting x-in. squares out of
each corner and folding up the sides. Find the
dimensions of the box that maximizes the
volume of the box
• HW: p.262 #1-59 AOO Try to get 6-7 more
correct
4.7 Optimization
Mon Dec 22
• Do Now – Optimization problem we
haven’t done yet
HW Review