Highway departments often store salt in sheds for use in road de-icing. Assume a

Question

Highway departments often store salt in sheds for use in road de-icing. Assume a conical shape as shown in the accompanying figure and assume that the shed is filled from the top of the structure. the circular base pad cost 50 dollars per square yard. The slanted sides of the cone are made of wood and shingles, and cost $30 dollars per square yard of curved surface. Each cone must hold 300 cubic yards of salt.

a. Write a mathematical programming model that will find the best height (h) of the cone and radius (r) of the base to minimize the total cost of the structure. Solve using calculus with substitution.

Explanation / Answer

cost of curved surface =30*pi*r*(r^2 +h^2)^0.5;

cost of base =50* (pi)*r^2 ;

volume of cone =(pi/3)*r^2*h =300

=>h = 900/(pi*r^2);

total cost =30*pi*r*(r^2+ 810000/(pi^2 *r^4) )^0.5 + 50* (pi)*r^2;

by differntiating the above expression and equating to zero we get

100*pi*r + 30*pi*(r^2+ 810000/(pi^2 *r^4) )^0.5 + 30 *pi*r*(r^2+ 810000/(pi^2 *r^4) )^(-0.5) *(2r +(810000/pi^2)(-4r^-5) ) =0;

solving we get r =k;

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