please help with finding the optimum radius, r Calculate this optimum radius r.The radius iscm.(Round to two decimal places as needed.) A soda can manufacturer wants to minimize the cost of the aluminum used to make their can. The can has to hold a volume V of soda.Assuming that the thickness of the can is the same everywhere, the amount of aluminum used to make the can will be proportional to itssurface area. That is, suppose the height of the can is h and the radius of the can is r, as shown in the figure on the right. Á soda can is acircular cylinder with radius r and height h. The curved surface area is 2arh and the area of each end cap is ar?. Then the manufacturerwants to minimize S = 2arh + 2ar? subject to the constraint that arh = V. Here we have used the formulas for the total surface area andvolume of a cylinder. Complete parts (a) through (d).(a) A real soda can has volume V = 279 cm. By substituting for h in the given equation, write S as a function of r only.558S(r) =+ 2r?(Type an expression.)(b) Describe the behavior of S(r) as r+oo.As r+0o, S(r)→ o0

As given in above statements that manufacture wants to minimize the surface area of the can i.e. S=2πrh+2πr2 And the subject to the constraint is: V=πr2h Where: r = Radius of the can h = Height of the cana) According to statement a real soda can has volume V=279 cm3 As we compare it to V=πr2h then the value of h is: 279=πr2hh=279πr2 Now the surface area of soda can when we put the value of h in it: S=558r+2πr2 .........1 So, according to first derivative test: S=558r+2πr2S'=-558r2+4πr The critical point of S' is: S'=0-558r2+4πr=0558r2=4πrr3=5584π r=5584π3r=3.54 cm As we know that the value of radius will not be negative and it should not be zero. So, the domain of r is: r∈0,∞ And as we draw the graph for it: So, as we put r=3.54 cm in equation (1) then the surface area is: S=558r+2πr2S=5583.54+2π3.542S=236.35 cm2 S, this is the minimum surface area for the soda can i.e. S=236.35 cm2 b) As r→∞ the surface area of soda can turn out to be more flatten and the shape of soda can changes from cylinder to disc type of soda can.