Question

A company plans to manufacture a rectangular box with a square base, an open top, and a volume of 404 cm3. The cost of the material for the base is 0.5 cents per square centimeter, and the cost of the material for the sides is 0.1 cents per square centimeter. Determine the dimensions of the box that will minimize the cost of manufacturing it. What is the minimum cost?

Answer #1

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A company plans to manufacture a rectangular container with a
square base, an open top, and a volume of 320 cm3. The cost of the
material for the base is 0.8 cents per square centimeter, and the
cost of the material for the sides is 0.2 cents per square
centimeter. Determine the dimensions of the container that will
minimize the cost of manufacturing it. What is the minimum
cost?

A rectangular box is to have a square base and a volume of 48
ft3. If the material for the base costs 4 cents per square foot,
material for the top costs 20 cents per square foot, and the
material for the sides costs 16 cents per square foot, determine
the dimensions of the square base (in feet) that minimize the total
cost of materials used in constructing the rectangular box.

A rectangular box is to have a square base and a volume of 45
ft3. If the material for the base costs 14 cents per square foot,
material for the top costs 6 cents per square foot, and the
material for the sides costs 6 cents per square foot, determine the
dimensions of the square base (in feet) that minimize the total
cost of materials used in constructing the rectangular box.

An open-top rectangular box is being constructed to hold a
volume of 300 in3. The base of the box is made from a
material costing 8 cents/in2. The front of the box must
be decorated, and will cost 12 cents/in2. The remainder
of the sides will cost 2 cents/in2.
Find the dimensions that will minimize the cost of constructing
this box.
Front width: _______ in.
Depth: ________ in.
Height: ________ in.

An open rectangular box (no top) is formed with a square base
and rectangular sides so that the total volume enclosed is 475 cu.
ft. What is the smallest amount of material (area) that can form
such a box?

A box with square base and open top is to have a volume of 10?3
. Material for the base costs $10 per square meter and material for
the sides costs $8 per square meter. Determine the dimensions of
the cheapest such container. Use the first or second derivative
test to verify that your answer is a minimum.

A rectangular box is to have a square base and a volume of 20
ft3. If the material for the base costs
$0.37/ft2, the material for the sides costs
$0.10/ft2, and the material for the top costs
$0.13/ft2, determine the dimensions (in ft) of the box
that can be constructed at minimum cost.

A 10 ft3 capacity rectangular box with open top is to be
constructed so that the length of the base of the box will be twice
as long as its width. The material for the bottom of the box costs
20 cents per square foot and the material for the sides of the box
costs 10 cents per square foot. Find the dimensions of the least
expensive box that can be constructed.

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 rectangular box is to have a square base and a volume of 20
ft3. If the material for the base costs $0.17/ft2, the material for
the sides costs $0.06/ft2, and the material for the top costs
$0.13/ft2,
(a) determine the dimensions (in ft) of the box that can be
constructed at minimum cost.
(b) Which theorem did you use to find the answer?

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