Question

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.

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 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?

Show the rectangular box with fixed volume V=27 and smallest
possible surface area is a cube. Hint: if the sides are x, y, and
z, then V=xyz and surface area is given by S=2xy+2yz+2xz. Use the
constraint in volume to turn the problem into one of two
variables.

A rectangular box with a square base has a volume of 4 cubic
feet. If x is the side length of the square base, and y is the
height of the box, find the total cost of the box as a function of
one variable The material for the bottom of the box costs $3 per
square foot, the top costs $2 per square foot, and the four sides
cost $5 per square foot. If x is the side length...

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...

A rectangular box with a square base has a volume of 4 cubic
feet. The material for the bottom of the box costs $3 per square
foot, the top costs $2 per square foot, and the four sides cost $5
per square foot.
(a) If x is the side length of the square base, and y is the
height of the box, find the total cost of the box as a function of
one variable.
(b) Find the critical number...

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

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?

A rectangular box with a square base has a volume of 4 cubic
feet. The material for the bottom of the box costs $3 per square
foot, the top costs $2 per square foot, and the four sides cost $5
per square foot Find the critical number of the cost function.

Suppose a tin box is to be constructed with a square base, an
open top and a volume of 32 cubic inches. The cost of the tin to
construct the box is $0.15 per square inch for the sides and $0.30
per square inch for the base.
1. The minimized cost of the tin box is A. $4.82 B. $3.50 C. $0
D. $9.07 E. none of the other answers
2. The cost is minimized at critical point x=a because...

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