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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.

Aug 11, 2018

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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   Aug 11, 2018