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Chapter 3: Applications of Differentiation
Section 3.8: Optimization
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Example 3.8.11
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A can in the shape of a right-circular cylinder, fashioned from a rectangle rolled into a cylinder and disks welded at the top and bottom, must have volume . What are the dimensions of the least-cost can if the side costs per square inch and the top and bottom cost per square inch?
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Solution
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Analysis
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Figure 3.8.11(a) animates rolling a rectangular sheet of material into a cylinder.
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If the radius of the resulting cylinder is , then the circumference (and hence, the width of the sheet) is .
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Of course, the height of the cylinder is the height of the rectangle, say, .
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The volume of the cylinder is then , which is a constraint on the objective function, itself the total cost of construction.
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>
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use plots, plottools in
ex3811:=module()
export F;
F:=proc(a)
local p1,p2;
p1:=plottools[polygon]([[cos(a),sin(a),0],[cos(a)-(2*Pi-a)*sin(a),sin(a)+(2*Pi-a)*cos(a),0],[cos(a)-(2*Pi-a)*sin(a),sin(a)+(2*Pi-a)*cos(a),3],[cos(a),sin(a),3]],color=green);
p2:=plot3d(1,theta=0..a,z=0..3,coords=cylindrical,color=red):
display(p1,p2,scaling=constrained);
end proc:
end module:
end use:
plots:-animate(ex3811:-F,[a],a=0..2*Pi,paraminfo=false,orientation=[-160,75,0],axes=none,lightmodel=none);
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Figure 3.8.11(a) Animation: rectangle to cylinder
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The objective function is the total cost, .
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Solve the constraint equation for so that the cost function is then or
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Analytic Solution
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Initialize and define the cost function
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Tools≻Load Package: Real Domain
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Loading RealDomain
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Control-drag
Context Panel: Assign Function
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Obtain the critical number
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Write and press the Enter key.
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Assign to a Name≻
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Unload the Real Domain package
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Tools≻Unload Package: Real Domain
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Unloading RealDomain
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Apply the Second-Derivative test
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Write
Context Panel: Evaluate and Display Inline
Context Panel: Evaluate at a Point≻
( ⇒ is a relative minimum)
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=
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Obtain the minimum cost
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Write and press the Enter key.
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Context Panel: Simplify≻Assuming Positive
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Define so that the dimensions minimizing cost are
= =
and
= = =
Figure 3.8.11(b) shows how the shape of a can with varies as the relative costs vary. For , the can looks like a mailing tube, so for the sake of the image all such values of are set to .
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=
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Figure 3.8.11(b) How relative cost affects shape of can
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