A cone is constructed by cutting a sector from a circular sheet of metal with ra

Question

A cone is constructed by cutting a sector from a circular sheet of metal with radius 14. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way. Write the objective function relating V and h in a form that does not include r. V = (Type an expression. Type an exact answer, using x as needed.) The interval of interest of the objective function is (Simplify your answer. Type your answer in interval notation.) The radius is r = and the height is h = Type exact answers, using radicals as needed.)

Explanation / Answer

Let sector subtend angle X at center of circle

Radius of cone is r

Hence, circumference of the base of cone is

C=2PI r=X*14

h^2+r^2=14^2

Volume, V=PIr^2h/3=PI(14^2-h^2)h/3

Volume must be positive hence

0<h<14

So interval of interest is (0,14)

V=PIr^2h/3=PI(14^2-h^2)h/3

V=14^2PIh/3-PIh^3/3

V'=14^2PI/3-PIh^2

V'=0 gives

h^2=14^2/3

h=14/sqrt{3}

h^2+r^2=14^2

14^2/3+r^2=14^2

r^2=2*14^2/3

r=14sqrt{2/3}

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