A metal storage tank with volume 500m^3 is to be constructed in the shape o a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal to construct the storage tank
well i cant figure it out ...do we need hemisphere on both side or u just planning to keep on one side
Let, radius = r meter, Volume = (4/3)pir^3 = 500 m^3, r^3 = 3*500/4pi, ===> r = (1500/4pi)^1/3 m surface area, S = (1/2)4pir^2 = 2pi[1500/4pi]^2/3, S = 104 sq.meter >========================< ANSWER
Assuming this is a vertical silo design with one hemisphere on top and that the metal needed also must include the base of the cylinder (it's not a dirt floor), then: ? ? V = π?r??h + ??π?r?, cylinder + hemisphere ? ? h = (V - ??π?r?) ? (π?r?), solving above for h ? ? A = 2π?r?h + 2π?r? + π?r? = 2π?r?h + 3π?r?, cylinder + hemisphere + base ? ? A = 2π?r?(V - ?π?r?) ? (π?r?) + 3π?r? ? ? A = 2?V ? r + ???π?r? ? ? dA ? dr = ????π?r - 2?V ? r? = 0 ? ? ????π?r? = 2?V ? ? r = ?[3V ? 5π] ≈ 4.5707815 m ? ? h = (V - ?π?r?) ? (π?r?) ≈ 4.5707815 m Interestingly, the r and h work out to the same value.
