a silo is to hold 1000 m^3 of corn. what dimensions will minimize the surface area?
You didn't say what shape the silo is to be. For a given volume, the minimum possible surface area is always a sphere. However, silos are usually cylindrical, so that's what I'll assume here. The volume of the silo = pi r^2 h where r is its radius and h is its height. So, pi r^2 h = 1000 Its surface area A = pi r^2 + 2 pi r h h = 1000 / (pi r^2) So A = pi r^2 + 2 pi r [1000 / (pi r^2)] = pi r^2 + 2000/r The minimum surface area is found by setting dA/dr = 0 dA/dr = 2 pi r - 2000/(r^2) So 2 pi r - 2000/(r^2) = 0 2 pi r = 2000/(r^2) r^3 = 1000/pi r = (1000/pi)^(1/3) h = 1000/(pi r^2) = (1000/pi)(r^3)^(-2/3) = (1000/pi)(1000/pi)^(-2/3) = (1000/pi)^(1/3) = r So, the surface area of the silo is minimised if r = h = (1000/pi)^(1/3) = 6.83 metres (approx.)
