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 axb, 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

What is the formula for 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. What do we need to know? COST What is the formula for cost? COST=60(Total Length of the Walls)

Therefore the formula for cost is? 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. Therefore the formula for cost is? C=60(6x+4y) How do you find the total length of the walls? L=6x+4y

REMEMBER: We want to have our formula in terms of one variable! 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. REMEMBER: We want to have our formula in terms of one variable! So what else does the problem tell us? What is the formula for area in terms of x and y? Area=600m2 =3xy

Solve for y (to get y in terms of x) Substitute for y in cost formula 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 y=200 x Solve for y (to get y in terms of x) Substitute for y in cost formula C=60(6x+4(200)) x

Do we have any restrictions on x or y? 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 Do we have any restrictions on x or y? YES - x0 and y0

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 3: Find the first derivative and find the value(s) of x when it is zero C=60(6x+4(200)) x C=360x+48000x-1 C’=360-48000x-2=360-48000 x2 0=360-48000 x2 360=48000 x2 360x2=48000 x11.547 x2133.3

Put in calculator first! What does the graph show you? 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? Step 4: If there is a restricted domain such as axb, 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. x m y m Put in calculator first! What does the graph show you? The end points are not going to be where cost is minimized, and where the derivative=0 is a min

What would the sign diagram look like? 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? Step 4: If there is a restricted domain such as axb, 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. x m y m What would the sign diagram look like? – + x11.547 Could see this by putting derivative in your calculator and looking at the table

But the question still hasn’t been answered… 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 But the question still hasn’t been answered… 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 or 36 cm2

YOU DO: 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?

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