# Z(t) represents the water height at time t; Z = 1m, then what is dz(t)/dt for 1m. Given: Radius (flask) : r(t) = 3.2/√(z+1) Bottom of flask, where z = 0, then r = 3.2 Time to escape shackles: 7...

Z(t) represents the water height at time t; Z = 1m, then what is dz(t)/dt for 1m.

Given:

Bottom of flask, where z = 0, then r = 3.2

Time to escape shackles: 7 minutes

Rate: 0.6π m3

Magician’s height: 1.82 meters

Volume of magician’s body: 0.53 m3

Height of cube: 0.5 m

0.6π m3 / minute x 7 minutes = 13.19 m3

Magician wants to be able to get out once the water reaches his head = Magician’s height + height of the block

Help from the teacher:

volume of the cone

V = (1/3)(pi)(r^2)h = (1/3)(pi)(z)[3.2/sqrt(z+1)]^2 = 3.41(pi)(z)/(z+1)

volume of the cube

Vc = 0.125 m^3

volume of the magician: Vm = 0.52 m^3

volume of the water:

Vw = V - (Vc + Vm)

the volume of the water in the flask can be written as a function of the height of the water above ground.

Z(t) represents the height of water above the ground level at time t. A formula for the rate of change of the height is a function of time or dz(t)/dt.

Z

dz(t)/dt

1m

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

The problem provides you a function of volume of water such that:

`V_w= V - (V_c + V_m)`

`V_w = 3.41 (pi*z)/(z+1) - 0.125 - 0.52`

`V_w = 3.41 (pi*z)/(z+1) - 0.645`

You need to differentiate the function of V_w with respect to t such that:

`(dV)/(dt) = (d(3.41 (pi*z)/(z+1) - 0.645))/(dz)*(dz)/(dt)`

`(dV)/(dt) = (3.41 (pi)(z+1) - 3.41 (pi*z))/((z+1)^2)*(dz)/(dt)`

`(dV)/(dt) = (3.41 pi*z + 3.41- 3.41 pi*z)/((z+1)^2)*(dz)/(dt)`

`(dV)/(dt) = (3.41)/((z+1)^2)*(dz)/(dt)`

The problem provides that `(dV)/(dt) = 0.6 pi ` hence, substituting 0.6 pi for (dV)/(dt) such that:

`0.6 pi = (3.41)/((z+1)^2)*(dz)/(dt)`

`(dz)/(dt) = (0.6 pi ((z+1)^2))/3.41`

You need to substitute 1 for z in equation `(dz)/(dt)`  such that:

`(dz)/(dt)= (0.6 pi ((1+1)^2))/3.41 =gt (dz)/(dt)~~3.141`

Hence, evaluating  `(dz)/(dt)`  for z =1m yields `(dz)/(dt)~~3.141.`