A symmetrical tank of height 15 feet, as shown in the figure on the next page, is leaking water through an aperture in its base. The aperture is a uniform circular hole with radius r_0 = 3 inches. The top and base radii of the tank are r_1 and r_2 respectively, where r_1 = r_2 = 2 feet. The curve passing through P1, P2 and P3 represents the arc of a parabola i.e. y = a + bx + cx^2 where a, b and c are suitable constants (to be found). The radius of the tank at P2 is r_3 and at P3 is r_4. It is also known that the vertical height of P2 from the top of the tank is h_4 such that h_4 = h_1/2.(adsbygoogle = window.adsbygoogle || []).push({});

a) Derive the height h of the water in the tank at any time t.

b) If the initial level of water in the tank is h_0, how long will it take to empty the whole tank?

c) How long will it take for half the water to leak out from the tank given the initial level h_0?

ASCII drawing of the tank:

************------------ <-- P1, h_1 = height from P1 to P3, rad.=r_1

************/********\

***********/**********\ This region is a parabolic curve outwards.

**********-************-

*********/**************\<-- P2, ht. h_4 = 0.5 * h_1, rad. = r_3

*******--****************--

******/********************\

***---**********************--- <-- P3, rad. = r_4, ht. from top h_1

***|*************************| ^

***|*************************| | <-- height is h_2

***|*************************| V

***\*************************/^

****\***********************/*| <-- height is h_3

*****\*********************/**| This region is a linear inwards.

******\*******************/***V

*******---------****---------- <-- bottom radius = r_2

**************^^^ radius of hole/aperture in bottom = r_0

The height values and radius values are:

h_0 = 10 feet

h_1 = 11 feet

h_2 = 2 feet

h_3 = 2 feet

r_3 = 3.5 feet

r_4 = 5 feet

Any help and guidance on this problem would be greatly appreciated.

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# Diff. Eq. Problem

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