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Find the volume of the water in the flask as a function of the height of the water...

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homeschool11 | (Level 1) Salutatorian

Posted January 27, 2012 at 4:12 AM via web

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Find the volume of the water in the flask as a function of the height of the water above the ground.

A magician’s feet will be shackled to a concrete cubic block placed on the bottom of a flask-shaped glass container. Radius of a flask, r, is a function of the height of the flask z, from the ground. `R(t) = 3.2/sqrt(z+1)` At the bottom of flask, z = 0 meters, so r = 3.2 meters. The container is filled with water at rate of `0.6pi m^3/min` . Magician knows he can escape the shackles in 7 minutes. He wants to get out of container when water level reaches top of his head, his height is 1.82 meters. What is the height of the concrete cubic block that he will stand on to make this happen?

Find the volume of the water in the flask as a function of the height of the water above the ground.

If the volume of the magician’s body is 0.52 m^3 and the height of the cube is 0.5m, what will be the volume of the water when it reaches the top of the magician’s head?

`Z(t) ` represents the height of water above the ground level at time t.

Derive a formula for the rate of change of the height as a function of time, or `d(z(t))/dt` ?

What does `(dz(t))/dt` represent?

 

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txmedteach | High School Teacher | (Level 3) Associate Educator

Posted January 27, 2012 at 8:10 AM (Answer #1)

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Let's parse this out. For the first bit, let's just assume the cube and magician don't matter.

Think about how we can find the volume of the flask given the height. Let's take it apart by disks, recognizing that the radius is given as a function of height. Now, given this radius, we can take a disc out of the flask that is infinitesimally small. Let's say that it's thickness (or height for purposes of the volume of said disc) will be `dz`. Well, we are basically separating the whole flask into a series of discs with radius `R(z)` and height `dz`. So, calculating the volume is simple. Just apply the formula for the volume of a cylinder:

`V_(cyl) = pi*R^2(z)*dz`

Recognizing that we want to add up all of the cylinders between the ground and height "z," we recognize that this is an integral. So, we integrate to find the total volume (let's call this `V(z)`):

`V(z) = int_0^z pi R^2(z) dz`

Substituting for our function R(z):

`V(z) = piint_0^z 3.2^2/(z+1) dz`

Now, we'll just solve the integral (recall `int (dx)/x = lnx`)

`V(z) = 10.24pi*(ln(z+1) - ln1) = 10.24pi ln(z+1)`

Now, we have our volume with respect to height (which is incidentally the answer to the second portion of the question). Let's now solve for z in terms of V so we can get z(t):
`V = 10.24piln(z+1)`

To start, let's divide both sides by `10.24pi`:

` ` `V/(10.24pi) = ln(z+1)`

Now allow both sides to be powers of e, and then you can get rid of the ln:

`e^(V/(10.24pi)) = z+1`

Now we have z in terms of V:

` ` `z = e^(V/(10.24pi)) - 1`

Ok, we're almost done with the first part. We know that the influx of water is at a constant rate (derivative) of `0.6pi` cubic meters/minute. As an equation:

`(dV)/(dt) = 0.6pi`

Therefore, the volume in the flask can be calculated by an integral (let's assume no water at time t=0):

`V(t) = int0.6pidt = 0.6pit`

Now, we can substitute this function in for V to get z(t):

`z(t) = e^((0.6pit)/(10.24pi)) - 1=e^(0.0586t)-1`

Haha, we're doing all of the problem before we even answer the first part!

Now, the magician wants to be able to get out once the water reaches his head, which will be at his height added to the height of the block (`1.82 + z_b` where `z_b` is the height of the block). The magician will get out in 7 minutes. Therefore, we get the following equation:

`1.82 + z_b = e^(0.0586*7) - 1`

Solving for `z_b` we get:

`z_b =-1.313m`

Well, this is interesting. In order for this problem to work out, the magician must dig a hole into the ground.

I looked at a lower-bound for the volume of the flask, where we take the radius at the man's height and used that to calculate the volume of a cylindrical section (notice the volume will be less than in the flask), and we get roughly 20 m^3. At the flow rate given, we only get 13 m^3 after 7 minutes. Clearly the flow rate is insufficient, and it tells me our negative block height is likely correct.

Now (running out of space), using our V(z) we can calculate the volume when the water reaches the top of the magician's head when he's on a box (`V_b` and `V_m` are the volumes of the box and magician):

`V(1.82+0.5)-V_b - V_m=10.24piln(3.32) - 0.645=37.957 m^3`

Finally, let's take the derivative of z(t) (`dotz`):

`dotz=0.0586e^(0.0586t)`

Yay! That was easy. Now what it means. This is the function giving the rate of change for the height of water over time. In other words, this tells you how fast the water is rising.

Hope that helps! I would have added more steps, but out of space.

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