Solution Found!
ight-circular cylinder standing on end is leaking water
Chapter 3, Problem 11E(choose chapter or problem)
Leaking Cylindrical Tank A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by
\(\frac{d h}{d t}=-\frac{A_{h}}{A_{w}} \sqrt{2 g h}\)
where Aw and Ah are the cross-sectional areas of the water and the hole, respectively.
(a) Solve the DE if the initial height of the water is H. By hand, sketch the graph of h(t) and give its interval I of definition in terms of the symbols \(A_{w}, A_{h}\), and H. Use \(g=32 \mathrm{ft} / \mathrm{s}^{2}\).
(b) Suppose the tank is 10 feet high and has radius 2 feet and the circular hole has radius inch. If the tank is initially full, how long will it take to empty?
Text Transcription:
\frac{d h}{d t}=-\frac{A_{h}}{A_{w}} \sqrt{2 g h}
Aw, Ah
g = 32 ft/s^2
Questions & Answers
QUESTION:
Leaking Cylindrical Tank A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by
\(\frac{d h}{d t}=-\frac{A_{h}}{A_{w}} \sqrt{2 g h}\)
where Aw and Ah are the cross-sectional areas of the water and the hole, respectively.
(a) Solve the DE if the initial height of the water is H. By hand, sketch the graph of h(t) and give its interval I of definition in terms of the symbols \(A_{w}, A_{h}\), and H. Use \(g=32 \mathrm{ft} / \mathrm{s}^{2}\).
(b) Suppose the tank is 10 feet high and has radius 2 feet and the circular hole has radius inch. If the tank is initially full, how long will it take to empty?
Text Transcription:
\frac{d h}{d t}=-\frac{A_{h}}{A_{w}} \sqrt{2 g h}
Aw, Ah
g = 32 ft/s^2
ANSWER:Step 1 of 4
a) We have a circular cylinder tank is leaking water through a hole in its bottom with a cross sectional area 
where 
and we have to obtain the amount of water in the cylinder at time 
This differential equation is a separable D.E, then we can solve it as
Then we have
