HI!!
Part[1] Find the dimensions of the box which minimize the amount of
materials used to construct the box.
Here you must find which dimensions minimize the Surface function:
You know that V = x^2*y = 1000 ,then y = 1000/x^2 , where x is the
length of the square side and y is the height of the box.
S = 2*x^2 + 4*x*y =
= 2*x^2 + 4*x*(1000/x^2) =
= 2*x^2 + 4000/x
S'(x) = 4*x - 4000/x^2 and S''(x)= 4 + 8000/x^3
To find where S(x) has a minimum we need to equal its first derivative to zero:
4*x - 4000/x^2 = 0 <==> 4*x = 4000/x^2 <==> x^3 = 1000 <==> x = 10 .
S''(10) = 4 + 8 = 12 > 0, then x = 10 is a minimum.
Then y = 1000/x^2 = 1000/100 = 10.
The dimensions of the box which minimize the amount of materials used
to construct it are 10" for each base side and 10" height (a cube!!).
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Part[2] The material used for the top piece costs twice as much per
square inch as the material used to construct the other piece of the
box. Find the dimensions of the box which minimizes the cost of
production.
Here we must repeat the procedure but minimizing the Cost function:
You know that V = x^2*y = 1000 ,then y = 1000/x^2 , where x is the
length of the square side and y is the height of the box.
C = x^2 + 4*x*y + 2*x^2 = (the last term shows that the top costs the
double than the base)
= 3*x^2 + 4*x*1000/x^2 =
= 3*x^2 + 4000/x
C'(x) = 6*x - 4000/x^2 and C''(x) = 6 + 8000/x^3
To find where C(x) has a minimum we need to equal its first derivative to zero:
6*x - 4000/x^2 = 0 <==> 6*x = 4000/x^2 <==> x = 8.7358
C''(8.7358) = 18 ,then x = 8.7358 is a minimum of the cost.
Then y = 1000/x^2 = 13.1037
The dimensions of the box which minimizes the cost of production are
8.7358 inches for each side of the base and 13.1037 inches of height.
The minimum cost is:
C(8.7358) = = 3*(8.7358)^2 + 4000/8.7358 = 686.8285
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Part[3] Your boss asks you if it would be better to have the shape of
the base, something besides a square. Repeat Part[2] with the base of
the box several different shapes (at least three.) Find at least one
other shape which would lower the cost of the box but still contain
1000 in3.
The shape of the base is a CIRCLE with radius r and the height of the box is h:
V = PI*r^2*h = 1000 ==> h = 1000/(PI*r^2)
Using the same method of Part 2:
C = 3*PI*r^2 + 2*PI*r*h =
= 3*PI*r^2 + 2*PI*r*1000/(PI*r^2) =
= 3*PI*r^2 + 2000/r
C'(r) = 6*PI*r - 2000/r^2 and C''(r) = 6*PI + 4000/r^3
6*PI*r - 2000/r^2 = 0 <==> 6*PI*r = 2000/r^2 <==> r^3 = 1000/(3*PI)
Then r = 4.73416
Since C''(4.73416) > 0 it is a minimum.
h = 1000/(PI*4.73416^2) = 44.6184
The minimum cost in this case is:
C(4.73416) = 3*PI*(4.73416)^2 + 2000/4.73416 = 633.6921
-----------------
The shape of the base is an EQUILATERAL TRIANGLE which each side
length is x and the height of the box is h:
V = sqrt(3)/4 * x^2 * h = 1000 ,then h = 4000/(sqrt(3)*x^2)
C = 3*sqrt(3)/4 * x^2 + 3*x*h =
= 3*sqrt(3)/4 * x^2 + 3*x*4000/(sqrt(3)*x^2) =
= 3*sqrt(3)/4 * x^2 + 3*4000/(sqrt(3)*x) =
= 3*sqrt(3)/4 * x^2 + sqrt(3)*4000/x =
C'(x) = 6*sqrt(3)/4 * x - sqrt(3)*4000/x^2
C''(x) = 6*sqrt(3)/4 + sqrt(3)*8000/x^3
6*sqrt(3)/4 * x - sqrt(3)*4000/x^2 = 0 <==>
<==> 6*sqrt(3)/4 * x = sqrt(3)*4000/x^2 <==>
<==> x^3 = 16000/6 <==> x = 13.8672
Since C''(13.8672) > 0 it is a minimum.
Then h = 4000/(sqrt(3)*13.8672^2) = 12.0094
The minimum cost in this case is:
C(13.8672) = 3*sqrt(3)/4 * (13.8672)^2 + sqrt(3)*4000/13.8672 =
= 749.4149
--------------------
The shape of the base is an HEXAGON which each side length is x and
the height of the box is h:
V = 3*sqrt(3)/2 * x^2 * h = 1000 ,then h = 2000/(3*sqrt(3)*x^2)
C = 3*3*sqrt(3)/2 * x^2 + 6*x*h =
= 9*sqrt(3)/2 * x^2 + 6*x*2000/(3*sqrt(3)*x^2) =
= 9*sqrt(3)/2 * x^2 + 4000/(sqrt(3)*x)
C'(x) = 9*sqrt(3) * x - 4000/(sqrt(3)*x^2)
C''(x) = 9*sqrt(3) + 8000/(sqrt(3)*x^3)
9*sqrt(3) * x - 4000/(4*sqrt(3)*x^2) = 0 <==>
<==> 9*sqrt(3) * x = 4000/(sqrt(3)*x^2) <==>
<==> x^3 = 4000/(9*sqrt(3)*sqrt(3)) <==> x^3 = 4000/27 <==>
<==> x = 5.2913
C''(5.2913) > 0 then x = 5.2913 is a minimum.
h = 2000/(3*sqrt(3)*5.2913^2) = 13.7473
The minimum cost inthis case is:
C = 9*sqrt(3)/2 * 5.2913^2 + 4000/(sqrt(3)*5.2913) =
= 654.6742
--------------------
I used the following pages for sources on area, surface and volume formulas:
"Area Formulas":
http://www.math.com/tables/geometry/areas.htm
"Surface Area Formulas":
http://www.math.com/tables/geometry/surfareas.htm
"Volume Formulas":
http://www.math.com/tables/geometry/volumes.htm
"Areas and Perimeters of Regular Polygons":
http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_AreaPerimeterRegularPolygons.xml
Search strategy:
area volume formulas
hexagon formula
I hope this helps you. If you find something unclear or some mistake
that I could did please feel free to use the clarification feature
before rate this answer. I will be glad to give you further assistance
on this if it is necessary.
Best regards,
livioflores-ga |