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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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...
(The entire section contains 655 words.)
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