PART 3

For our part II project, we choose the Hilsburg non alcoholic malt drink conture glass bottle.Thus, we have to convert the line of the bottle into several functions so we can calculate the cross section and the volume of the bottles. Below are the lines of the bottle converted into cartesian plan.

The reason we only includes half of the bottle body because it will be easier to calculate its volume using the integration. And to calculate the cross section area of the bottle, we just have to multiply by 2 to have the overall value of its cross section area.

So, we have to names all the lines as f(x)_1, f(x)_2, f(x)_3, f(x)_4 and f(x)_5.
And we have to find the value of each functions.The coordinates for each functions are stated below:

f(x)_1 : (23,3) ,(10.5,3)
f(x)_2 : (10.5, 3) , (9, 1.75)
f(x)_3 : (9, 1.75) , (1, 1.25)
f(x)_4 : (1, 0) , (1, 1.25)
f(x)_5 : (23, 0) , ( 23, 3)

And the value of the functions are :

f(x)_1 : y = 3 ; where ( 23 <= x <= 10.5 )
f(x)_2 : 12y = 14x – 27 ; where ( 10.5 <= x <= 9 )
f(x)_3 : 16y = x + 7 ; where ( 9 <= x <= 1 )
f(x)_4 : x = 1 ; where (0 <= y <= 1.25 )
f(x)_5 : x = 23 ; where ( 0 <= y <= 3 )

Thus, we can calculate the area under the graph using the information above.

A = 10.5 ʃ 23 f(x)_1 + 9 ʃ 10.5 f(x)_2 + 1 ʃ 9 f(x)_3
A = 10.5 ʃ 23 (3 dx) + 9 ʃ 10.5 (14x – 27 dx) + 1 ʃ 9 (x + 7 dx )
   = 10.5[3x]23 + 9[7x2 – 27x]10.5 + 1[x2/2 + 7x]9
   = (37.5) + (164.25) + (96)
A = 297.75

And because we have to multiply A by 2, so the area of cross section of the bottle are
Cross-section bottle,
   = A*2
   = 595.5 cm2
And to find the volume of the bottle, we use the volume revolution of integration

Vb = π 10.5 ʃ 23 [f(x)_1]2 + π 9 ʃ 10.5 [f(x)_2]2 + π 1 ʃ 9 [f(x)_3]2
Vb = π10.5 ʃ 23 (3 dx)2 + π9 ʃ 10.5 (14x – 27 dx)2 + π1 ʃ 9 (x + 7 dx )2
      = π10.5 ʃ 23 (9 dx) + π9 ʃ 10.5 (196x2 – 756x + 729 dx) + π1 ʃ 9 (x2 +14x +49 dx )
Vb = 19347.67 π cm3