Consider a right circular cone of base radius 4 and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surface resting on the base of the cone. Find the largest possible total surface area (in sq cm) of the cylinder.	Pick
A) 100π/3 B) 80π/3 C) 120π/7 D) 130π/9 E) 110π/7
Explanation :   Let the radius of cylinder be r and its height is h.   So, the height remained above cylinder is 10-h   Also, triangles ABC and ADE are similar. so ratio of corresponding sides must be equal   i.e AB/AD = BC/DE.   i.e. (10-h)/10 = r/4.   so 40 - 4h = 10r (by cross multiply)   h = (40-10r)/4 = 10-2.5r   Now we have total surface area of cyl = 2*π*r*h + 2*π*r2.   Putting value of h = 10 - 2.5r here, we get S = 2*π*r*(10-2.5r) + 2*π*r2.   S = 20*π*r- 5 * π * r2 + 2 * π * r2 .   S = 20 *π*r- 3*π*r .   Now, we have to maximize S, so differentiate it w.r.t r, we get   S = 20*π - 6 * π * r setting it to zero, we get   =&gt; 20*π - 6 * π * r= 0   =&gt; r = 20/6 = 10/3.   So, h = 10 - 2.5r = 10 - 2.5*10/3 = 5/3   So,   =&gt; S = 2 * π * r * h + 2 * π * r2.   =&gt; S = 2 * π * 10/3 * 5/3 + 2 * π* 100/9.   =&gt; S = 2 * π * 50/9 + 2 * π* 100/9.   =&gt; S = 2π150/9.   =&gt; S = 100π/3.   100π/3nswer : 100π/3
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Consider a right circular cone of base radius 4 and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surface resting on the base of the cone. Find the largest possible total surface area (in sq cm) of the cylinder.