A water tank, open at the top, consists of a right circular cylinder and right circular cone. If the altitude of the cylinder is 3 times its radius, and the altitude of the cone is 2 times the same radius, find the number of square meter of sheet required to construct a tank having a capacity of 4000 liters.
First 4000 L = [ 4000 * 1 / 1000 m^3 ] = 4 m^3
We need to find the radius
The volume of the cylinder = pi *r^2 * heght = pi * r^2 * 3r = 3 pi * r^3 m ^3
If the cone had the same height as the cylinder, it would hold 1/3 as much water as the cylinder....bit since its height is 2/3 that of the cylinder....its volume is (1/3)(2/3) = 2/9 that of the cylinder = (2/9)(3 pi r^3 = (2/3) pi r^3 m^3
So...he total volume is [ 3 + 2/3] pi r^3 m^3 = [ 11/3 pi r^3 ] m ^3
Set this equal to 4 m^3
11/3 pi r^3 = 4 divide both sides by 11/3 pi
r^3 = 4 / [ 11/3 pi ] take the cube root of both sides
r ≈ .7 m
To find the approximate amount of sheet metal needed, we need to find the lateral area of the the side of the cylinder and cone
Lateral area of the cylinder = 2 pi * radius * height = 2 pi * (.7m) (3* .7m) = 2pi * 1.47 ≈ 2.94 pi m^2
We next need to find the slant height of the cone = √ [ radius^2 + height ^2 ] = √ [ .7^2 + 1.4^2 ] ≈ 1.57 m
So....the lateral area of the cone is pi * radius * slant height = pi * .7m * 1.57m ≈ 1.1 pi m^2
So.....the total amount of sheet metal need is ≈ [2.94 + 1.1] pi m^2 = 4.04 pi m^2
