D.E. Problem Explained: Limits and Units

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[SOLVED] D.E. problem

Homework Statement


See the attachment.

Homework Equations


The Attempt at a Solution


dh/dt = 1/100 - 1/4h

Why is the limit not h = 1/25 or 4 cubic feet?
 

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Please be careful about parentheses! I had first interpreted "1/4h" as "1/(4h)" and got a silly answer!

However, your equation is wrong. Since water comes in at 1 cubic foot per second and the "water tower" has a uniform cross section area of 100 square feet, yes, that causes the water level to rise at 1/100 feet per second which are the correct units for dh/dt.

But the water is also going out at (1/4)t cubic feet per second which is NOT the correct units. If water is going out at (1/4)t cubic feet per second, how fast is the water level dropping in feet per second?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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