Question

A manufacturer sends you 100m2 of material to construct a box (single layer, closed top). The box must have a square base and be of maximum volume. Let sbe side length the base of the box, and hthe height of the box. a) Write an equation for the surface area covered by the material. b) Determine a formula for the volume V as a function of the side of s only. c) Determine the dimensions such that of the box has maximum volume.

Answer #1

A box with a square base and open top must have a volume of
108000 cm^3. We wish to find the dimensions of the box that
minimize the amount of material used.
First, find a formula for the surface area of the box in terms of
only x, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in
terms of x.]
Simplify your formula as much as possible....

A box with a square base and open top must have a volume of
157216 cm3cm3. We wish to find the dimensions of the box that
minimize the amount of material used.
First, find a formula for the surface area of the box in terms of
only xx, the length of one side of the square base.
[Hint: use the volume formula to express the height of the box in
terms of xx.]
Simplify your formula as much as possible....

A closed box with a square base is to have a volume of
2000in2. The material for the top and bottom of the box
is to cost $6 per in2, and the material for the sides is
to cost $3 per in2. If the cost of the material is to be
the least, find the dimensions of the box. Prove/justify
your answer.

A box with an open top has a square base and four sides of equal
height. The volume of the box is 225 ft cubed. The height is 4 ft
greater than both the length and the width. If the surface area is
205 ft squared. what are the dimensions of the box?
What is the width of the box?.
What is the length of the box?

A rectangular box is to have a square base and a volume of 16
ft3. If the material for the base costs
$0.14/ft2, the material for the sides costs
$0.06/ft2, and the material for the top costs
$0.10/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.

A manufacturer wants to design an open box, i.e, a box without a
lid. The box has a square base and a surface area of 108 square
inches. What dimensions will produce a box with minimum volume.

A rectangular storage container with an open top has a volume of
10 m3 . The length of the base is twice its width. Material for the
base costs $10 per sqaure meter and material for the sides costs $6
per square meter. (a) Find an equation for the volume of the box,
relating the variables of the height of the box and the width of
the base of the box. (b) Use the previous equation to solve for the...

A rectangular box must have a volume of 2 cubic meters. The
material for the base and top costs $ 2 per square meter. The
material for the vertical sides costs $ 8 per square meter. (a)
Express the total cost of the box in terms of the length (l) and
width (w) of the base. C = $ (b) Find the dimensions of the box
that costs least. length = meters width = meters height =
meters

1. A six-sided box has a square base and a surface area of 54
m^2. Let V denote the volume of the box, and let x denote the
length of one of the sides of the base. Find a formula for V in
terms of x.
2. What is the maximum possible volume of the box in Problem 1?
Note that 0< x≤3√3.

A carpenter wants to construct a closed-topped box whose base
length is 2 times the base width. The wood used to build the top
and bottom costs $7 per square foot, and the wood used to build the
sides costs $6 per square foot. The box must have a volume of 12
cubic feet. What equation could be used to find the smallest
possible cost for the box?

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