Optimization Lesson

​Optimization Lesson

​"Distractions lead to Subtractions from your life."

By: Mr. C.

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Multiple Choice

What can you conclude from this number line? #1

• there is a maximum for A at 12

• there is a minimum for A at 12

• you will have a profitable year

• the stars are aligning nicely

Multiple Choice

If y=2x-8, what is the minimum value of the product xy? #8

• -16
• -8
• -4
• 2

Multiple Choice

Find two positive numbers such that the sum of the first number and five times the second number is 40, and their product is maximum. #10

• 25 and 3

• 15 and 5

• 20 and 4

• 35 and 1

• Cannot be determined

Multiple Choice

Find two real numbers whose difference is 500 and whose product is a maximum. #19

• 250 and 250

• 400 and -100

• 150and -350

• 250 and -250

• Cannot be determined

Multiple Choice

Find two real numbers whose difference is 500 and whose product is a minimum. #20

• 250 and 250

• 300 and -200

• 150and -350

• 250 and -250

• Cannot be determined

Multiple Choice

A rectangle has an area of 81 cm^2.

What is the maximum perimeter possible? #23

• 810cm

• 72 cm

• 18cm

• 36 cm

• cannot be determined

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Multiple Choice

Farmer Jo has 32 square feet of land in which to make an enclosure for bunnies, chicks, and penguins. (see picture)

Choose the equation that represents this information. #2

• $$2x+4y=32$$

• $$2x+2y=32$$

• $$xy=32$$

• $$A=3xy$$

Multiple Choice

same question, now solve it and answer the question:

Farmer Jo has 32 square feet of land in which to make an enclosure for bunnies, chicks, and penguins.  There will be fencing put around the entire enclosure and in the middle (as pictured) to create 3 sections, what dimensions should the overall enclosure be to use the least amount of fence? #3

• The overall enclosure should be 8 feet by 8 feet

• The overall enclosure should be 8 feet by 4 feet

• The overall enclosure should be 32 square feet

• The overall enclosure should be  $$\sqrt{128}=8\sqrt{2}$$  feet on each dimension

Multiple Choice

A rectangle is bounded by the x-axis and the parabola  $$y=12-x^2$$ .  What length and width should the rectangle have so that its area is a maximum?  

Given the constraint equation above and the optimization equation  $$A=2xy$$  , choose the DERIVATIVE of the merged (combined) equation. #6

• $$A'=2xy$$  

• $$A'=\ \ 24-6x^2$$  

• $$A'=2x\left(12-x^2\right)$$  

• $$A'=12-2x$$  

Multiple Choice

What is the process of optimization in calculus? #15

• The process of optimization in calculus involves finding the maximum or minimum value of a function within a given domain.

• The process of optimization in calculus involves finding the average value of a function within a given domain.

• The process of optimization in calculus involves finding the integral of a function within a given domain.

• The process of optimization in calculus involves finding the derivative of a function within a given domain.

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Multiple Choice

A farmer wants to construct a rectangular pigpen using 400 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area? #7

• 100ft x 200ft

• 102ft x 196 ft

• 50 ft x 300 ft

• 50 ft x 175 ft

Multiple Choice

Pick the correct interpretation from the given number line for V', the derivative of V. #9

• V has a minimum at 3

• V has a maximum at both 1 and 5

• V has a maximum at 0 and a minimum at 6

• V has a maximum at 1 and a minimum at 5

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Multiple Choice

You want to make a box to contain dirt and your pet earthworm. Using a 7 in by 10 in a rectangle of cardboard, you cut congruent squares from the corners and fold up the sides.

Choose the equation would you use to do Calculus to find the maximum volume of dirt (including worms) the box can hold. #13

• $$V=\left(7-2x\right)\left(10-2x\right)$$

• $$V=x\left(7-2x\right)\left(10-2x\right)$$

• $$V=x\left(7-x\right)\left(10-x\right)$$

• $$V=x\left(7+2x\right)\left(10+2x\right)$$

Multiple Choice

A closed rectangular shipping box with a square base is to be made from 120 square inches of cardboard. What dimensions should the box be for maximum volume?

Choose the constraint and optimization equations that represent the problem. #5

• $$x^2=120$$  and  $$V=x^3$$  

• $$2x+y=120$$  and   $$V=xy$$  

• $$x^2y=120$$  and  $$V=2x^2+4xy$$  

• $$2x^2+4xy=120$$  and  $$V=x^2y$$  

Multiple Choice

The volume of a cylindrical tin can with a top and a bottom is to be 16π cubic inches. If a minimum amount of tin is to be used to construct the can, what must be the height, in inches, of the can? #16

• 2 ³√2
• 2√2
• 4
• 2 ³√4