Optimization

In optimization problems we are trying to find the maximum or

minimum value of a variable. The solution is called the optimum

solution.

Optimization Problem Solving Method

Step 1: If possible, draw a large, clear diagram. Sometimes more than one diagram is necessary

Optimization Problem Solving Method

Step 2: Construct an equation with the variable to be optimized (maximized or minimized) as the subject of the formula(the y in your calculator) in terms of one convenient variable, x. Find any restrictions there may be on x.

Optimization Problem Solving Method

Step 3: Find the first derivative and find the value(s) of x when it is zero

Optimization Problem Solving Method

Step 4: If there is a restricted domain such as axb, the maximum/minimum value of the function may occur either when the derivative is zero or at x=a or at x=b. Show by a sign diagram that you have a maximum or minimum situation.

A industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The walls, internal and

external, will cost $60 per meter to build. What dimensions would the shed have to minimize the

cost of the walls?

Step 1: If possible, draw a large, clear diagram. Sometimes more than one diagram is necessary.

x m

y m

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

Step 2: Construct an equation with the variable to be optimized as the subject of the formula in terms of one convenient variable, x. Find any restrictions there may be on x.

COST

COST=60(Total Length of the Walls)

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

Step 2: Construct an equation with the variable to be optimized as the subject of the formula in terms of one convenient variable, x. Find any restrictions there may be on x.

L=6x+4y

C=60(6x+4y)

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

Step 2: Construct an equation with the variable to be optimized as the subject of the formula in terms of one convenient variable, x. Find any restrictions there may be on x.

Area=600m2 =3xy

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

Step 2: Construct an equation with the variable to be optimized as the subject of the formula in terms of one convenient variable, x. Find any restrictions there may be on x.

Area=600m2=3xy

Solve for y (to get y in terms of x)

y=200 x

Substitute for y in cost formula

C=60(6x+4(200)) x

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

Step 2: Construct an equation with the variable to be optimized as the subject of the formula in terms of one convenient variable, x. Find any restrictions there may be on x.

C=60(6x+4(200)) x

YES - x0 and y0

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

C=60(6x+4(200)) x

Step 3: Find the first derivative and find the value(s) of x when it is zero

C=360x+48000x-1

C’=360-48000x-2=360-48000 x2

0=360-48000 x2

360=48000 x2

360x2=48000

x2133.3x11.547

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

Step 4: If there is a restricted domain such as axb, the max/min value of the function may occur either when the derivative is 0 or at x=a or at x=b. Show by a sign diagram that you have a max or min.

The end points are not going to be where cost is minimized, and where the derivative=0 is a min

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

x11.547

Step 4: If there is a restricted domain such as axb, the max/min value of the function may occur either when the derivative is 0 or at x=a or at x=b. Show by a sign diagram that you have a max or min.

+ –

Could see this by putting derivative in your calculator and looking at the table

An industrial shed is to have a total floor space of 600 m2 and is to be divided into 3 rectangular rooms of equal size. The

walls, internal and external, will cost $60 per meter to build. What dimensions would the shed have to minimize the cost of

the walls?x m

y m

What are the dimensions?

3x meters by y m

3x meters by (200/x) m

3(11.547) meters by (200/11.547) m

34.6 meters by 17.3 m

An open rectangular box has square base and a fixed outer surface area of 108 cm2. What size

must the base be for maximum volume?

ANSWER:6 cm by 6 cm

or36 cm2

Solve using derivatives: Square corners are cut from a piece of 20 cm by 42 cm cardboard which is then bent into the form of an open box. What size squares should be removed if the volume is

to be maximized?

YOU DO:

TICKET OUT

A closed box has a square base of side x and height h.

Write down an expression for the volume, V, of the box

Write down an expression for the total surface area, A, of the box

The volume of the box is 1000 cm3

Express h is terms of x

Write down the formula for total surface area in terms of x

Find the derivative (dA/dx)

Calculate the value of x that gives a minimum surface area

Find the surface area for this value of x