Houdini plans to have his feet shackled on the top of a concrete block...

[prefreshmanstudent]

I'm a pretty average joe in my calculus class and I can't figure out this problem no matter how hard I try. I found the solution to the same problem with a different set of numbers: http://courses.ncssm.edu/math/POW/POW 09_10/Calculus Challenge #9 SOLUTION.pdf
This is the problem:
Houdini plans to have his feet shackled on the top of a concrete block which was placed on the bottom of a giant flask. The cross-sectional radius of the flask, measured in feet, is given as a function of the height y from the ground by the formula
r(h)=10/(sqrt(h+1))
with the bottom of the flask at y = 0 foot. The flask is to be filled with water at a constant rate of 22π cubic feet per minute. Houdini’s job is to escape the shackles before he drowns! Houdini knows that he can escape the shackles in exactly 10 minutes. For dramatic effect, he wants to escape at the moment the water level reaches the top of his head. Houdini is 6 feet tall. In the design of the apparatus, Houdini can change only the height of the concrete block on which he stands.
1.How high should the block be? (you will have to find the volume of the water when the water level reaches he top of Houdini's head. Express the volume of the water in the flask as a function of the height of the liquid above the ground level.
2.Houdini wanted to be able to know his progress of his escape. Find the equation for dh/dt as a function of h(t) itself. Use this equation to find out how fast the water is rising when the flask first begins to fill. How fast is the water rising when it reaches the top of his head?

What I've done so far:

If you know how to do it you can you also provide work? Thanks so much.

 

[stapel]

If you know how to do it you can you also provide work?

Sorry, but that's not how this forum works. For further information, kindly please re-read the "Read Before Posting" announcement.

I'm a pretty average joe in my calculus class and I can't figure out this problem no matter how hard I try. I found the solution to the same problem with a different set of numbers: http://courses.ncssm.edu/math/POW/POW 09_10/Calculus Challenge #9 SOLUTION.pdf

Actually, they've used different variables, etc, to. You seem not to have understood the set-up (perhaps because they gave it to you, instead of having you figure out the logic yourself?) which is confusing things.

Houdini plans to have his feet shackled on the top of a concrete block which was placed on the bottom of a giant flask. The cross-sectional radius r of the flask, measured in feet, is given as a function of the height h from the ground by the formula:

. . . . .\(\displaystyle r(h)\, =\, \dfrac{10}{\sqrt{\strut h\, +\, 1\,}}\)

with the bottom of the flask at h = 0 foot.

The variable is important! Do you see how, in changing the height at the bottom, they had to change the form of the denominator? Be sure that you do!

The flask is to be filled with water at a constant rate of 22pi cubic feet per minute.

Houdini’s job is to escape the shackles before he drowns! Houdini knows that he can escape the shackles in exactly 10 minutes. For dramatic effect, he wants to escape at the moment the water level reaches the top of his head. Houdini is 6 feet tall. In the design of the apparatus, Houdini can change only the height of the concrete block on which he stands.

1. How high should the block be? (you will have to find the volume of the water when the water level reaches he top of Houdini's head. Express the volume of the water in the flask as a function of the height of the liquid above the ground level.
2. Houdini wanted to be able to know his progress of his escape. Find the equation for dh/dt as a function of h(t) itself. Use this equation to find out how fast the water is rising when the flask first begins to fill. How fast is the water rising when it reaches the top of his head?

What I've done so far:

\(\displaystyle r(h)\, =\, \dfrac{10}{\sqrt{\strut h\, +\, 1\,}}\)

\(\displaystyle A(h)\, =\, \pi \left(r(h)\right)^2\)

Would it be correct to assume that "A(h)" means "cross-sectional area of slices of the volume", and that you're going to follow the lead of the worked solution at the link by working with volumes of revolution?

\(\displaystyle V(h)\, =\, \)\(\displaystyle \displaystyle \int\, A(h)\, dh\, \)

. . .\(\displaystyle \displaystyle =\, \int\, \pi\, \)\(\displaystyle \left(\dfrac{10}{\sqrt{\strut h\, +\, 1\,}}\right)^2\, dh\)

. . .\(\displaystyle \displaystyle =\, \pi\, \int\, \)\(\displaystyle \dfrac{100}{h\, +\, 1}\, dh\)

. . .\(\displaystyle \displaystyle =\, 100\, \pi\, \int\, \)\(\displaystyle \dfrac{1}{h\, +\, 1}\, dh\)

. . .\(\displaystyle =\, 100\, \pi\, \ln\lvert\, h\, +\, 1\, \rvert\, +\, C\)

\(\displaystyle \dfrac{dV}{dt}\, =\, 22\, \pi\)

\(\displaystyle \mbox{At }\, t\, =\, 10, \, \mbox{ the volume of the flask is }\, 220\, \pi.\)

\(\displaystyle \mbox{We have to find an interval of }\, h\, \mbox{ for which }\, \Delta\, h\, =\, 6\, \mbox{ for a volume of }\, 220\, \pi.\)

Previous to the last line above, you seem to have been mimicking what the worked solution provided. But I do not understand what you mean by "finding an interval"...?

Please reply with your reasoning. Thank you!