A worker is to construct an open rectangular box with a square base and a volume of 48 ft3. If material for the bottom costs​ $6/ft2 and material for the sides costs ​$74/ft2 what dimensions will result in the least expensive​ box? What is the minimum​ cost?Let x be the length of one side of the base of the box. Express the total cost of the​ materials, C, in terms of x.

Question
Asked Nov 7, 2019

A worker is to construct an open rectangular box with a square base and a volume of 48 ft3. If material for the bottom costs​ $6/ftand material for the sides costs ​$74/ftwhat dimensions will result in the least expensive​ box? What is the minimum​ cost?

Let x be the length of one side of the base of the box. Express the total cost of the​ materials, C, in terms of x.

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Expert Answer

Step 1

Note: Please check the material cost for side. Because the dimension seems to be unrealistic since 74 is a large number. 

Given that a worker is to construct an open rectangular box with a square base.

The volume of 48 ft3 and if material for the bottom costs $6/ft2 and material
for the all the sides costs $74/ft2
To find: the dimensions that will result in the least expensive box and the
minimum cost
help_outline

Image Transcriptionclose

The volume of 48 ft3 and if material for the bottom costs $6/ft2 and material for the all the sides costs $74/ft2 To find: the dimensions that will result in the least expensive box and the minimum cost

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Step 2

To begin with, 

Let x be the length of one side of the base of the box. Let y be the height of the
box. Express the total cost of the materials, C, in terms of x.
We know that the volume of the box is:
Volume
height = x* x*y x2y
length * breadth
Now, volume of the box is 48 ft3
48
x2y
48 implies y
.(1)
х2
help_outline

Image Transcriptionclose

Let x be the length of one side of the base of the box. Let y be the height of the box. Express the total cost of the materials, C, in terms of x. We know that the volume of the box is: Volume height = x* x*y x2y length * breadth Now, volume of the box is 48 ft3 48 x2y 48 implies y .(1) х2

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Step 3

Since, the box is an open rectangle, the surface area of the...

48
= x+
x2
192
Surface area x2 + 4xy = x2 + 4x
.2(from 1)
х
Material for the bottom costs $6/ft2 and material for the all the sides costs
$74/ft2
Therefore, the total cost of the material is:
192
14208
C 6 x2 74
бх? +
.. (from 2)
X
X
help_outline

Image Transcriptionclose

48 = x+ x2 192 Surface area x2 + 4xy = x2 + 4x .2(from 1) х Material for the bottom costs $6/ft2 and material for the all the sides costs $74/ft2 Therefore, the total cost of the material is: 192 14208 C 6 x2 74 бх? + .. (from 2) X X

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Calculus

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