Optimization problems

Multiple Choice

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

Choose the equation would you use in order to do Calculus to find the maximum volume of dirt (including worm) the box can hold?

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

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

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

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

Multiple Choice

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.

• $$2x+4y=32$$

• $$2x+2y=32$$

• $$xy=32$$

• $$A=3xy$$

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 in around the entire enclosure and in the middle (as pictured) to create 3 sections, what dimensions should the overall enclosure be in order to use the least amount of fence?

• 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

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

• 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

Multiple Choice

$$A'=\frac{-2\left(x-12\right)\left(x+12\right)}{x^2}$$  and x has to be positive.
What work makes sense to happen next?

• plug in 12 and -12, resulting in a maximum

• plug in 10 and 14, resulting in a minimum

• plug in 10 and 14, resulting in a maximum

• take the derivative of the expression

Multiple Choice

A closed rectangular shipping box with 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. 

• $$x^2=120$$  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

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.

• $$A'=2xy$$  

• $$A'=\ \ 24-6x^2$$  

• $$A'=2x\left(12-x^2\right)$$  

• $$A'=12-2x$$  

Fill in the Blank

If y= -3x + 24, what is the MAXIMUM VALUE of the product xy?

type in the answer below, no space.

Multiple Choice

We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field so we won't need any fencing on that side. Determine the dimensions of the field that will enclose the largest area.

• x=250 ft, y=125 ft

• x=150 ft, y=200 ft

• x=125 ft, y=100 ft

• x=200 ft, y=150 ft