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
subsitute that for h in the other equation
SAcost=2pirh+12pir^2
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=![\sqrt[3]{\frac{150}{7 \pi}}](https://tex.z-dn.net/?f=%5Csqrt%5B3%5D%7B%5Cfrac%7B150%7D%7B7%20%5Cpi%7D%7D)
by subsituteion we find that the value of h will be then![h=\frac{4(7 \pi -1)\sqrt[3]{7350}}{21\sqrt[3]{\pi}}](https://tex.z-dn.net/?f=h%3D%5Cfrac%7B4%287%20%5Cpi%20-1%29%5Csqrt%5B3%5D%7B7350%7D%7D%7B21%5Csqrt%5B3%5D%7B%5Cpi%7D%7D)
aproximately
the radius should be about 1.8964m and the height should be about 53.0789m
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
subsitute that for h in the other equation
SAcost=2pirh+12pir^2
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=
by subsituteion we find that the value of h will be then
aproximately
the radius should be about 1.8964m and the height should be about 53.0789m