Question

I can’t solve this help please?!

Answer 1

Height of the cylinder `h= "its volume"/"area of the circular base"`

`=(108pi)/(pix^2)=108/x^2` cm

Total surface area of the solid

`A="Surface area of cylinder"+ "Surface area of the cone"`

`=>A=(2pix*h+pix^2)+pix*3x`

`=>A=2pix*108/x^2+4pix^2=4pix^2+(216pi)/x" "cm^2`

Differentiating w r to x we get

`(dA)/(dx)=8pix-(216pi)/x^2`

For minimum value of `A`, we have `(dA)/(dx)=0`

So `=8pix-(216pi)/x^2=0`

`=>x^3=216/8=27=3^3`

Hence `x=3` cm

So minimum value of surface area

`A_"min"=4pi*3^2+(216pi)/3=108pi` `cm^2`

Answer 2

`"see explanation"`

Explanation:

`(a)`

`"using the following formulae"`

`• " curved surface of cylinder "=2pirh`

`"where r is the radius of the base and h the height"`

`• " curved surface of cone "=pirs`

`"where r is the radius of the base and s the slant height"`

`"surface area "="area of base + curved surface of cylinder/cone"`

`=pir^2+2pirh+pirs`

`"we require to find h using the given volume"`

`V_("cylinder")=pir^2h=108pi`

`rArrh=(108pi)/(pir^2)`

`"substitute "r=x`

`A=pix^2+(2pix xx(108pi)/(pix^2))+pixx x xx3x`

`color(white)(SA)=pix^2+(216pi)/x+3pix^2`

`color(white)(SA)=4pix^2+(216pi)/x`

`(b)`

`"using "color(blue)"differential calculus"`

`A=4pix^2+216pix^-1`

`rArrA'=8pix-216pix^-2`

`rArrA''=8pi+432pix^-3`

`"for max/min set "A'=0`

`rArr8pix-(216pi)/x^2=0larrcolor(blue)"multiply by "x^2`

`rArr8pix^3-216pi=0`

`rArrx^3=(216pi)/(8pi)rArrx=root(3)(216/(8))=3`

`"to determine if this produces a minimum area"`

`"use the "color(red)"second derivative test"`

`• " if "A''>0" then minimum"`

`• " if "A''<0" then maximum"`

`A''(3)=8pi+432/(3)^3>0" hence minimum when "x=3`

`rArr"minimum area "=4pi(3)^2+(216pi)/3=339.29`