A grain silo consists of acylinder on a cone. The radius ofthe cylinder is 1.5 m. The slantheight of the cone is 2.5 m. Theoverall height of the grain silo is20 m.Find:(i) The height of the cone.(ii) The height of the cylinder.(iii) The capacity of the silo(when full) in terms of pi.(iv) The depth of the grain (asmeasured from the bottom-mostpoint of the cone) when the silois one seventh full.
cylinder radius = 1.5m slant height of cone = sqrt{r^2+h^2} = 2.5m Overall height of silo = 20m (i)From slant height formula, sqrt{r^2 + h^2} = 2.5 =sqrt{1.5^2 + h^2} = 2.5 1.5^2 + h^2 = 2.5^2 2.25 + h^2 = 6.25 h^2 = 6.25 - 2.25 =4 h = sqrt{4} =2 Now you got height of cone from axis = 2m. (ii)Total height of grain silo = 20m Total height = cylinder height + cone height from axis Cylinder height = Total height - cone height from axis = 20 - 2 =18m (iii)Total capacity of silo = Capacity of cylinder + Capacity of cone = Cylinder volume = cone volume Cylinder volume = pi(r^2)(h) Cone volume = 1/3(pi)(r^2)(h) Cylinder volume = pi(1.5^2)(18) =(2.25)(18)pi =(40.5pi) cubic meter Cone volume = 1/3(pi)(1.5^2)(2) =1/3(2.25)(2)pi =1/3(4.5)pi =(1.5pi) cubic meter Total capacity = 40.5pi + 1.5pi =(42pi) cubic meter [Substitute pi with constant 3.142 if required] (iv)Total capacity = 42pi Capacity when one seventh full = 1/7(42pi) =7pi Does 7pi fill the entire cone since cone is bottom? 7pi - total capacity of cone = 7pi - 1.5pi = 5.5pi(above zero, so whole cone is filled) Height of cylinder filled = volume / (pi multiply r^2) 5.5pi / (pi)(1.5^2) = 5.5pi / 2.25pi =22/9 Depth = height of cone + filled cylinder height = 2 + 22/9 = 18/9 + 22/9 = 40/9 m [Simplify fraction to 4 4/9 or 4.444444 will also be good.]
