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exis [7]
2 years ago
5

An open box with a square base is constructed so that it has a volume of 64in^3. The base needs to be made out of a material tha

t costs twice as much as the sides. What are the dimensions of the box, what are the dimensions of the box x and y (x being the width and depth of the square dimensions, and y being the height of the box) which will minimize the cost of the box?
Mathematics
1 answer:
Ivahew [28]2 years ago
4 0

The volume of a box is the amount of space in the box

The dimensions that minimize the cost of the box is 4 in by 4 in  by 4 in

<h3>How to determine the dimensions that minimize the cost</h3>

The dimensions of the box are:

Width = x

Depth = y

So, the volume (V) is:

V = x^2y

The volume is given as 64 cubic inches.

So, we have:

x^2y = 64

Make y the subject

y = \frac{64}{x^2}

The surface area of the box is calculated as:

A =x^2 + 4xy

The cost is:

C = 2x^2 + 4xy --- the base is twice as expensive as the sides

Substitute y = \frac{64}{x^2}

C = 2x^2 + 4x * \frac{64}{x^2}

C =2x^2 +  \frac{256}{x}

Differentiate

C' =4x -  \frac{256}{x^2}

Set to 0

4x -  \frac{256}{x^2} = 0

Multiply through by x^2

4x^3 -  256= 0

Divide through by 4

x^3 -  64= 0

Add 64 to both sides

x^3 =  64

Take the cube roots of both sides

x = 4

Recall that:

y = \frac{64}{x^2}

So, we have:

y = \frac{64}{4^2}

y = 4

Hence, the dimensions that minimize the cost of the box is 4 in by 4 in  by 4 in

Read more about volume at:

brainly.com/question/1972490

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Solve the inequality t/4 &gt;7<br> And 2c/3 +1&gt;7
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Answer:   c > 12

Step-by-step explanation:tep  1  :

           c

Simplify   —

           3

Equation at the end of step  1  :

       c            

 ((2 • —) -  1) -  7  > 0  

       3            

Step  2  :

Rewriting the whole as an Equivalent Fraction :

2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

        1     1 • 3

   1 =  —  =  —————

        1       3  

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

2.2       Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

2c - (3)     2c - 3

————————  =  ——————

   3           3    

Equation at the end of step  2  :

 (2c - 3)    

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Step  3  :

Rewriting the whole as an Equivalent Fraction :

3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

        7     7 • 3

   7 =  —  =  —————

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Adding fractions that have a common denominator :

3.2       Adding up the two equivalent fractions

(2c-3) - (7 • 3)     2c - 24

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Step  4  :

Pulling out like terms :

4.1     Pull out like factors :

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Equation at the end of step  4  :

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Step  5  :

5.1    Multiply both sides by  3  

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Solve Basic Inequality :

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Inequality Plot :

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One solution was found :

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