(See attached file for full problem description with diagram)

An open-top box is to be made as follows: squares of a certain size will be cut away from each of the four corners of a 20" x 30" rectangle, and the ends will be folded upward to form the corner seams, as shown. How big should the square cutouts be in order to maximize the volume of the resulting box?

#2:
Let f(x) = ((x-2)2) / (1+x)

a. Use derivatives to identify all local maxima and/or minima of f, if any.
b. Use calculus to find the highest and lowest values attained by the function f on
the interval 0 (greater than or equal to) x (less than or equal to) 5.

#3:
Let r(x) = (2/3)x3 - x - (1/3)x4 + (1/20)x5 + 1 :
Find the second derivative and set it equal to zero, in order to find inflection points.

#4:
Sketch continuous functions g and h satisfying the given conditions. Please sketch carefully enough to show the concavity clearly, and label axes and tickmarks where appropriate:

h ' (x) is negative for x < 1 and h' (x) is positive for x > 1; h" (x) is positive for
0 < x < 2, and h" (x) < 0 elsewhere

An open-top box is to be made as follows: squares of a certain size will be cut away from each of the four corners of a 20" x 30" rectangle, and the ends will be folded upward to form the corner seams, as shown. How big should the square cutouts be in order to maximize the volume of the resulting box?

Answer:
The height of the box is
The width of the box is 20-2x
The length of the box is 30-2x
The volume of the box is
In order to find the max volume, the first derivative will be taken

When , the x and V values could be max.

is discarded since 10" max can be cut out.

So will ...

Solution Summary

This solution is comprised of a detailed explanation to use derivatives to identify all local maxima and/or minima of f, if any.

An open top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting out equal squares at each corner and then folding up the flaps. Let x denote the length of each side of the square to be cut out.
Find the function V that represents the volume of the box in terms of x. Show and explain the answer, and a

1.You are to design a container box by cutting out the four corners of a square cardboard sheet that is 1600 cm2 in area. The box must have a square base and an open top. Determine the dimensions of the box that give maximum volume.
2.Sketch the graph of the function f(x)=x2+4 Identify the following features of the graph:

An open box with a square base is required to have a volume of 27 cubic feet. Express the amount A of material that is needed to make such a box as a function of the length x of a side of the square base. I have A(27/x^2) +x^3....i have no clue to if this is correct.

If 2400 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Volume = cubic centimeters.

U-Pack-Em sells cardboard boxes for the do-it-yourself mover. Their most popular size has a volume of 2
cubic feet. As shown in the figure below, the top and bottom are made using four flaps. The price of
cardboard is $0.20/ft'. What are the raw materials cost and dimensions for the cheapest box that can be
manufactured? (Be

BLOCK WITH A FORCE
THERE IS A BLOCK OF SIDE X METRES RESTING ON A FRICTIONLESS SURFACE
A FORCE F IS APPLIED AT THE LEFT TOP OF THE BOX(TAKE THE DIRECTION OF FORCE IS FROM LEFT TO RIGHT) HORIZONTALLY
NOTE: THE FORCE IS NOT AT THE CENTRE OF GRAVITY.
PROVE THAT THE BOX TENDS TO ROTATE ABOUT THE CENTRE OF GRA

Consider an open-topbox with a square base and a volume of 108 cubic inches. Let x be the length of a side of the base.
a) Calculate the height h as a function of x. Is this function even, odd, or neither?
b) What is the domain of the function above? (Note that there may be physical and/or mathematical restrictions.)