SOLUTION: A box with a square bottom and no top is to be made to contain 100 cubic inches. Bottom material costs five cents per square and side material costs two cents per square inch. Find

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Question 1155393: A box with a square bottom and no top is to be made to contain 100 cubic inches. Bottom material costs five cents per square and side material costs two cents per square inch. Find the cost of least expensive box that can be made.



Please, help out me out.

Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.

Let x be the size of the squared base edge and h be the height.


Then the volume is  x^2*h = 100  cubic inches 

and the lateral surface area is 4*xh square inches.


The total cost is  C(x,y) = 5x^2 + 2*4*xh = 5x^2 + 8xh.


So we need to minimize  C(x,y) = 5x^2 + 8xh  under restriction x^2*h = 100.

We then express h =  and substitute it into the expression for C(x,y).


We then get

    C(x,y) =  +  =  + ,

and we need to minimize this function.


Find the derivative and equate it to zero

    10x -  = 0.


From this equation, find x

    10*x^3 = 800

    x^3 =  = 80.

    x =  = 4.309  inches.

Then h =  =  = 5.386 inches.


ANSWER.  The base size is 4.309 inches;  the height is 5.386 inches.

Solved.

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If you want to see many other similar solved problems, look into the lesson
    - Calculus optimization problems
in this site.


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