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Instructions: Please complete the post project problem in the steps indicated. Clearly show your work in detail for credit.

A rectangular container with open top is required to have a volume of 24 cubic meters. Also, one side of the rectangular base is required to be 4 meters long. If material for the base costs $8 per square meter, and material for the side’s costs $2 per square meter, find the dimensions of the container so that the cost of material to make it will be a minimum.
(a) Make a diagram

(b)What formulas will be used in this problem?

(c) Figure out what the constraint is

(d) What do you want to maximize or minimize?
Find the Critical points.
a) Test the critical points.  b) State any standard results or theorems used

Step 6: Answer the question

 

Step 7: Write a brief summary report of the problem and the solution obtained above (something like an elevator pitch of the problem and the solution)

 May 4, 2021
edited by Guest  May 4, 2021
edited by Guest  May 4, 2021
 #1
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+1

A rectangular container with open top is required to have a volume of 24 cubic meters. Also, one side of the rectangular base is required to be 4 meters long. If material for the base costs \(8\ per\ square\ meter,\ and\ material\ for\ the\ side’s\ costs \) 2 per square meter, find the dimensions of the container so that the cost of material to make it will be a minimum.

 

Hello Guest!

 

V = abh = 24

h = x

a = 4

4bx = 24

b = 6/x

 

\(Costs = f(x)=ab\times 8 +2(ax+bx)\times 2\\ f(x)=4\cdot\frac{6}{x}\cdot 8 +2(4x+\frac{6}{x}\cdot x)\cdot 2\\ x\in \mathbb R|x=0\\ f(x)=\frac{192}{x}+16x+24\)

 

\(\frac{df(x)}{dx}=-\frac{192}{x^2}+16=0\\ 16x^2=192\\ x^2=12\\ \color{blue}x=3.464\\\)

When the cost is minimal, the rectangular container has a height of 3.464.

You can do the rest of the task yourself.

laugh  !

asinus

 May 4, 2021
edited by asinus  May 4, 2021
edited by asinus  May 4, 2021
 #2
avatar+2127 
+1

Hello :))

 

I'm a bit confused on how f(x) = 192/x +16x + 24 went to the second part. 

What does d stand for?

 

Thanks

=^._.^=

catmg  May 4, 2021
 #3
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+1

Is this asinus?

 

 

nvm it is why does it show him as guest?

Guest May 4, 2021
edited by Guest  May 4, 2021
 #4
avatar+2127 
0

Sometimes accounts aren't logged in. 

 

=^._.^=

catmg  May 4, 2021
 #5
avatar+11870 
+1

Hello catmg,

 

\(\frac{df(x)}{dx}\)  is the differential quotient of function  \( f(x)=\frac{192}{x}+16x+24\)

If this is equal to zero, the function f (x) has an extreme at this point.

greetings

laugh  !

asinus  May 4, 2021
 #6
avatar+11870 
0

Hello Guest #3,

I wasn't logged in

laugh  !

asinus  May 4, 2021

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