4.7 Applied OptimizationWed Jan 14

Do Now

Differentiate

1) A(x) = x(20 - x)

2) f(x) = x^3 - 3x^2 + 6x - 12

HW Review

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 optimization problems

Optimization

• 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 12in is bent into the shape of a rectangle. Which dimensions produce the rectangle of maximum area?

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 10 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

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

• HW: p.262 #1 3 5 7 23 35

4.7 OptimizationThurs Jan 15

• 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 3 5 7 23 35

• 1) a) y = 3/2 - x

b) A = 3/2 x - x^2 c) [0, 3/2]

d) A = 0.5625 with x = y = 3/4• 3) Let circumference of circle be 5.28• 5) middle of wire• 7) x = 300/(1 + pi/4) y = 150/(1 + pi/4)• 23) x = 22.36m y = 44.72 m• 35) each compartment is 600 x 400m

Example 7.3

• Find the point on the parabola

closest to the point (3, 9)

More Practice

• (green book) worksheet p.306 #7-10 all 13 19 20

Closure

• Journal Entry: How can we use derivatives to find an optimal design to a situation?

• HW: worksheet p.306 #7-10 all, 13 19 20• Take home test Tues

3.7 Optimization Tues Jan 20

• Do Now• A 216 sq. meter rectangular corn field is to be

enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for whole field will require the smallest length of fence?

HW Review: worksheet 7-10 15 19 20

• 7) 30’ x 60’, perimeter = 120 ft

• 8) 24ft x 48ft, max area = 1152 sq ft

• 9) 20 ft x 30 ft

• 10) 20 ft x 40 ft

• 13) (0, 1)

• 19) about 1.2137

• 20) about 2.263 in

Box Project (extra credit)

• Box Project

• Due Wednesday Jan 22

• The work must be included– On separate sheet or on box

Minimizing the cost

• When we want to minimize cost, we want to differentiate a cost function.

• We may need a 2nd equation to substitute variables

Ex 1• We want to construct a box whose base length is 3

times the base width.  The material used to build the top and bottom cost $10/ft2 and the material used to build the sides cost $6/ft2.  If the box must have a volume of 50ft3 determine the dimensions that will minimize the cost to build the box.

Ex 2

• Farmer Al needs to fence in 800 square yards, with one wall being made of stone which costs $24 per yard, and the other three sides being wire mesh which costs $8 per yard. What dimensions will minimize the cost?

Ex 3You are constructing a set of rectangular pens in which

to breed hamsters. The overall area you are working with is 60 square feet, and you want to divide the area up into

six pens of equal size as shown below. The cost of the outside fencing is $10 a foot. The inside fencing costs $5 a foot. Find the dimensions that would minimize the cost

Closure

• What is the hardest part when solving an optimization problem? Why? What is the most important part?

• HW: Ch 4 Take Home Test• Ch 4 Take Home Test due tomorrow

HW Review: worksheet p.318 #3-5 11 20 21

• 3) 250 x 500 ft

• 4) 500 x 500 ft

• 5) 500 x 750 ft

• 11) 40 x 80 ft

• 20) 15 x 15 x 10

• 21) 10 x 10 x 20