wait what?
removing the flat surfaces...
... oh!
I get it
okey dokey
basicaly find the shape that gives you a volume of 200 cubic meters and and minimizes the amount of materials needed to make the conainer
find the volume
we can take the 2 hemispheres and combine them to get 1 sphere
we can take the cylinder as well
vsphere=(4/3)pir^3
vcylinder=hpir^2
so total volume would be (4/3)pir^3+hpir^2
now find surface area which is the amount of sheet metal we will need to construct it
surface area=lateral area of cylinder+surface area of sphere
lateral area of cylinder=2pirh
surface aea of sphere=4pir^2
total surface area=2pirh+4pir^2
but, the hemishpere part costs 3 times more
so just say the cylinder part costs 1 dollar per square meter and hemisphere part costs 3 dollars so we mutiply hemishpere part by 3 to get
SAcost=2pirh+12pir^2
so we got
v=200=hpir^2+4/3pir^3
solve for h since the r exponents are tricky
[tex]h=\frac{200-\frac{4}{3} \pi r^3}{\pi r^2}[/tex]
subsitute that for h in the other equation
SAcost=2pirh+12pir^2
[tex]SAcost=2 \pi r (\frac{200-\frac{4}{3}\pi r^3}{\pi r^2})+12 \pi r^2[/tex]
now we simpilify and take the derivitive to find the value of r where SAcost is a minimum
we get at that the derivitive of SAcost is 0 at r=[tex]\sqrt[3]{\frac{150}{7 \pi}}[/tex]
by subsituteion we find that the value of h will be then [tex]h=\frac{4(7 \pi -1)\sqrt[3]{7350}}{21\sqrt[3]{\pi}}[/tex]
aproximately
the radius should be about 1.8964m and the height should be about 53.0789m
