2.  

Packaging

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 16 in. long and 6 in. wide, find the dimensions (in inches) of the box that will yield the maximum volume. (Round your answers to two decimal places if necessary.)

smallest value=? in ?in largest value =?in

3.

Minimizing Packaging Costs

A rectangular box is to have a square base and a volume of 36 ft³. If the material for the base costs $0.24/ft², the material for the sides costs $0.15/ft², and the material for the top costs $0.16/ft², determine the dimensions (in ft) of the box that can be constructed at minimum cost. (Refer to the figure below.)

A closed rectangular box has a length of x, a width of x, and a height of y.

x= ?ft   y=? ft

4.

Book Design

A book designer has decided that the pages of a book should have 1 in. margins at the top and bottom and one half in in. margins on the sides. She further stipulated that each page should have an area of 98 in.² (see the accompanying figure). Determine the page dimensions that will result in the maximum printed area on the page.

x= ?in y= ?in

7.

Racetrack Design

The accompanying figure depicts a racetrack with ends that are semicircular in shape. The length of the track is 1056 ft (1/5 mi). Find l and r such that the area of the rectangular region of the racetrack is as large as possible. (Round your answers to the nearest foot.)

r=? ft   l=? ft

What is the area enclosed by the track in this case? (Round your answer to the nearest square foot.)

? ft²

Please answer each question. Thank you

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x³ + y³ − 3x² − 9y² − 9x

1. Find the local maxima of the function:

(1) f(x,y) = xy, subject to the constraint that x+y-1=0. Result should be 1/4.

2. Find the local minima of the functions:

(1) f(x,y) = x^2+y^2, subject to the constraint that xy-3=0. Result should be 6.

(2) f(x,y) = x^2+4xy+y^2, subject to the constraint that x-y-6=0. Result should be -18.

In a typical optimization problem (max/min problem), we want to find a relative maximum or relative minimum of a function. Our process is to

• find the derivative of the function,

• set that derivative equal to zero,

• and solve for x.

Use complete sentences to explain why this process makes sense.

1) You drop a pebble into a well. 10 seconds later, you hear the sound of the pebble hitting the water. How deep is the well? For what part of the 10 seconds was the pebble falling? How long did it take the sound to travel back up the well? What assumptions did you make in solving the problem?

2) Suppose another well is just the depth at which the time it takes your pebble to hit the water is the same as the time it takes the sound of hitting the water to return to the top. How deep is the well?

3) Suppose another well is just the depth at which the time it takes your pebble to hit bottom is 15 times greater than the time it takes the sound of it hitting return to the top? How deep is the well?

I need a differential equations approach to the problem please.

Derive expressions for the transverse modulus and the longitudinal modulus of a 3-
component composite

A tank originally contains 100 gal of fresh water. Then water containing 1 2 lb of salt per gallon is poured into the tank at a rate of 2 gal/min, and the mixture is allowed to leave at the same rate. After 10 min the process is stopped, and fresh water is poured into the tank at a rate of 4 gal/min, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional 10 min. (Round your answer to two decimal places.)

Determine all the inflection points of g(x) = 3x² - x³

^{a) Only (1,2)}

b) Both (0,0) and (2,4)

c) only (2,4)

d) All of (0,0), (1,2), and (2,4)

e) only (0,0)

(Optimization) A cylindrical container w/ circular base is to hold 64 in3. Find the dimension so that the amount of metal required is a minimum when the container is a) an open cup and b) closed can.

Determine the area of ​​the largest isosceles triangle that can be written into the unit circle.

It is advisable to let the corners in the triangle be given by (0.1), (x, y) and (−x, y), all on the unit circle, with x ≥ 0 but where we allow y to be negative.

How to find the local extrema for y^3- 3yx^2 -3y^2-3x^2+1. 

2. Sketch a function which satisfies both conditions OR state that there is no such example.

(a) Differentiable at x=0 AND not continuous at x=0

(b) Not differentiable at x=0 AND continuous at x=0

What does it mean for a function to be/not be differentiable at x=0? What does it mean for a function to be/not be continuous at x=0?