# Calculus project: volumes, rates, etc

• UWMpanther
In summary, the problem involves Houdini being trapped in a giant flask with his feet shackled to a block. Water is being pumped into the flask at a rate of 22pi. The goal is to calculate the height of the block so that the water reaches the top of Houdini's head in 10 minutes. By setting up the equation v=100pi(ln(h+1)) and solving for h, it is found that the block should be approximately 8 feet tall. Additionally, Houdini has derived an equation for the rate of change of the water level, dh/dt, as a function of h(t). The water level changes at a rate of 0.22(h+1)-1 when the

## Homework Statement

Houdini is in a giant flask and he stands on a block where his feet are shackled and you need to calculate some vital information to help him out.

Cross section of flask = r(h)=10/(sqrt(h+1))
water is being pumped into the flask at 22pi
Takes Houdini 10 mins to escape
Ignore blocks volume and his volume
He's 6ft tall

1) You first task is to find out high the block should be so that the water reaches the top of his head at the 10 min mark. Express the water in the flask as a function of the height of the liquid above ground level. I calculated this to be approximately 2ft.

220pi=100pi(ln(h+1))
h≈8ft

2) 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 derived 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 Houdini's Head? Express h(t) as a function of time.

## Homework Equations

v=100pi(ln(h+1))
dv/dt=100pi((h'+1)/(h+1))

## The Attempt at a Solution

solve for h'? h'=.22(h+1) - 1
integrate this?

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I stopped reading after "1) You first task is to find out high the block should be so that the water reaches the top". I suggest you write the problem correctly indicating what each variable mean (ex: what is h? and what does 22pi means?) and write grammatically correct sentences.