5) Find the metric tons of water displaced by the submarine drawn below (up to t

Question

5) Find the metric tons of water displaced by the submarine drawn below (up to the conning tower) using the given gridlines and the specifications as stated. Assume the body of the sub was created using a curve and then generating a solid of revolution by spinning it around the central axis. To find that equation, use the scale diagram below with the overall length of the shaded region given as 420 ft. Keep in mind it will be easier to do this if you flip the picture vertical and place the "Vertex" at each tip of the sub. The region below the conning tower can be assumed to be cylindrical. 3

Explanation / Answer

5) from the given diagram, the submarine can be divided into four parts

the cylinder in the top

the quadratic cross section on the left and the right and the cylinder in the center

let the volumes of these parts be V1, V2, V3, V4 respectively

then

let 1 unit of the grid represent 1 unit in calculation

V1 = pi*(4)^2*5

V4 = pi*(6)^2*9

for V3 and V2

Volume inside a paraboloid is given by

dV = kx^2*dx*2*pi*x = 2*pi*k*x^3*dx

V = pi*y^2/2k

hene for V3

y = 22

also

y = 22 when x = 7

hence

22 = k(7^2)

k = 22/49

V3 = pi*(22^2)*49/2*22 = 1693.3184

similiarly

for V4

y = 11, when x = 7

hence

1 1= k(7^2)

k = 11/49

V4 = pi*11^2*49/22 = 846.659220142

hence

V = V1 + V2 + V3 + V4 = 3809.1810521 cubic units

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