Does anyone know how to solve or how to begin to solve the following problem?
Houdini had his feet tied to the top of a block which was placed on the bottom of a giant laboratory flask. The radius of the flask, in ft, was given as a function of height z from the ground by the formula
r(z)=(10)/(square-root(z))
with the bottom of the flask at z=1 ft. The flask was filled with water at a rate of 22 cubic ft/min.
Houdini knew it would take ten minutes to escape the shackles. He wanted to escape at the moment the water level reached the top of his head. Houdini was exactly 6 feet tall.
In the design of the apparatus, he was allowed to specify 1 thing: the height of the block.
Your task is to find out how high this block should be. Express the volume of water in the flask as a function of the height of the liquid above the ground level. What is the volume when the water level reaches the top of Houdini's head? (Neglect Houdini's volume and the volume of the block)
2007-05-29
07:46:24
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1 answers
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asked by
jgk5252
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Science & Mathematics
➔ Mathematics
Let h(t) be the height of the water above ground level at time t. In order to check the progress of his escape moment by moment, Houdini derives the equation for the rate of change dh/dt as a function of h(t) itself. Derive this equation. How fast is the water level changing when the flask first starts to fill? How fast is it changing when the water just reaches the top of his head? Express h(t) as a function of time.
Houdini would like to be able to perform this trick with any flask. Help him plan his next trick by generalizing the derivation of part (b). Consider a flask with cross sectional radius r(z) (an arbitrary function of z) and a constant inflow rate dV(t)/dt = A. Find dh/dt as a function of h(t).
2007-05-29
07:46:43 ·
update #1