Torricelli's law states that the rate at which water drains out of a tank through a hole in the bottom is proportional to the square root of the water level. In other words, if V(t) is the volume of water in the tank at time t, and h(t) is the height (depth) of the water at time t, then: dV dt = k√h As is, this differential equation is hard to work with since it involves both h(t) and V(t). dV dV dh We can use chain rule to rewrite the derivative: dt dh dt dV dh dh dt . dh dt = k√h = . 3 seconds. SO If we can express volume as a function of the water height, then we can find a formula for in terms of h, and we'll be left with a differential equation only involving ʼn and t. dV dh Suppose we have a simple cylindrical tank with a hole in the bottom. The tank has height 14 and radius 5. Find a formula for the volume of water in the tank when the water height is h and use it to set up the differential equation for the height of water in the tank as it drains. Your equation should involve h and k. B) Using the differential equation from part A, suppose the tank starts filled with water. After 6 seconds, the height in the tank is 13. How long does it take the tank to drain (measured from when the water first started draining)? C) Now suppose we have a different cone-shaped tank with a hole in the bottom. The height of the cone is 14 and the radius at the top is 5. Find a formula for the volume of water in the tank when the water height is ʼn and
Torricelli's law states that the rate at which water drains out of a tank through a hole in the bottom is proportional to the square root of the water level. In other words, if V(t) is the volume of water in the tank at time t, and h(t) is the height (depth) of the water at time t, then: dV dt = k√h As is, this differential equation is hard to work with since it involves both h(t) and V(t). dV dV dh We can use chain rule to rewrite the derivative: dt dh dt dV dh dh dt . dh dt = k√h = . 3 seconds. SO If we can express volume as a function of the water height, then we can find a formula for in terms of h, and we'll be left with a differential equation only involving ʼn and t. dV dh Suppose we have a simple cylindrical tank with a hole in the bottom. The tank has height 14 and radius 5. Find a formula for the volume of water in the tank when the water height is h and use it to set up the differential equation for the height of water in the tank as it drains. Your equation should involve h and k. B) Using the differential equation from part A, suppose the tank starts filled with water. After 6 seconds, the height in the tank is 13. How long does it take the tank to drain (measured from when the water first started draining)? C) Now suppose we have a different cone-shaped tank with a hole in the bottom. The height of the cone is 14 and the radius at the top is 5. Find a formula for the volume of water in the tank when the water height is ʼn and
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can't get this one right. I could use an explanation please.
Transcribed Image Text:Torricelli's law states that the rate at which water drains out of a tank through a hole in the bottom is
proportional to the square root of the water level. In other words, if V(t) is the volume of water in the tank at
time t, and h(t) is the height (depth) of the water at time t, then:
dV
dt
As is, this differential equation is hard to work with since it involves both h(t) and V(t).
dV
dV
dh
We can use chain rule to rewrite the derivative:
dt
dh
dt
= k√h
dV dh = k√h
dh dt
dh
dt
If we can express volume as a function of the water height, then we can find a formula for in terms of h,
and we'll be left with a differential equation only involving h and t .
"
Suppose we have a simple cylindrical tank with a hole in the bottom. The tank has height 14 and radius 5. Find
a formula for the volume of water in the tank when the water height is h and use it to set up the differential
equation for the height of water in the tank as it drains. Your equation should involve and k.
dh
dt
SO
dV
dh
B) Using the differential equation from part A, suppose the tank starts filled with water. After 6 seconds, the
height in the tank is 13. How long does it take the tank to drain (measured from when the water first started
draining)?
seconds.
||
C) Now suppose we have a different cone-shaped tank with a hole in the bottom. The height of the cone is 14
and the radius at the top is 5. Find a formula for the volume of water in the tank when the water height is h and
use it to set up the differential equation for the height of water in the tank as it drains. Your equation should
involve h and k.
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