Calc 2 -Toricelli help please!

[Daved26]

The shape is a cylinder. the height of the water is .1 meters. The radius of the drain hole is .0015m. The radius of the cylinder itself is .0635 meters. I have already solved using Using symbolic techniques i must find an actual formula for V(t) using the DEQ dV/dt= -a*SQRT(2gh) where a is the area of the drain hole (pi*.0015^2) and h is the height in meters from the hole to the surface of the liquid. I believe i have solved for V(t) correctly but i am not sure I have:
V = pi*r^2*h, so h = V/pi*r^2

We can then write the equation as:

dV/dt = -pi*r^2*sqrt(2*g*V/pi*r^2) = -sqrt(2*g*V*pi*r^2)

This is a separable equation:

dv/sqrt(V) = -sqrt(2*g*pi*r^2) dt

Integrate:

2*sqrt(V) = c -sqrt(2*g*pi*r^2)*t

where c is the constant of integration.

sqrt(V) = C - sqrt((g*pi*r^2)/2)*t

where C = c/2 is just another way of writing the constant.

V(t) = [C - sqrt((g*pi*r^2)/2)*t]^2

If, at time t = 0, V(0) = Vo, then:

Vo = C^2

C = sqrt(Vo)

so:

V(t) = [sqrt(Vo) - sqrt((g*pi*r^2)/2)*t]^2

After this i am supposed to form a table of values in excel but when i do i do not seem to get the right values. I put time in the a column and this equation in the b column and A1 or time for t and 9.8 for g. any suggestions or help would be greatly appreciated

 

[DrPhil]

Using symbolic techniques i must find an actual formula for V(t) using the DEQ dV/dt= -a*SQRT(2gh) where a is the area of the drain hole (pi*.0015^2) and h is the height in meters from the hole to the surface of the liquid. I believe i have solved for V(t) correctly but i am not sure I have:
V = a*h, so h = V/a

I fear a vital geometric feature is missing. What is the actual shape of the container? The way you have done it, the container is a tiny cylinder, giving you V = a*h. Usually the tank will be a large cylinder (of known Radius), or a cone (with known slope). Can you check to see if that additional information is given in the question?

Your work may be correct if the shape is a cylinder, but use V=(pi R^2)*h instead of a*h. If the shave is a cylinder the form will involve h^3.

THANK YOU for showing your work! - please correct the Volume and repost.

 

[Daved26]

I hope i fixed all that you were asking.

 

[DrPhil]

The shape is a cylinder. the height of the water is .1 meters. The radius of the drain hole is r=.0015m. The radius of the cylinder itself is R= .0635 meters. I have already solved using Using symbolic techniques i must find an actual formula for V(t) using the DEQ dV/dt= -a*SQRT(2gh) where a is the area of the drain hole (pi*.0015^2) and h is the height in meters from the hole to the surface of the liquid. I believe i have solved for V(t) correctly but i am not sure I have:
V = pi*R^2*h, so h = V/pi*R^2

We can then write the equation as:

dV/dt = -pi*r^2*sqrt(2*g*V/pi*R^2) = -sqrt(2*g*V*pi*r^2) NO

This is a separable equation:

dv/sqrt(V) = -sqrt(2*g*pi*r^2) dt *(r/R)

Integrate:

2*sqrt(V) = c -sqrt(2*g*pi*r^2)*t *(r/R)

This is a separable equation:

dv/sqrt(V) = -sqrt(2*g*pi*r^2) dt *(r/R)

This is a separable equation:

dv/sqrt(V) = -sqrt(2*g*pi*r^2) dt *(r/R)


where c is the constant of integration.

sqrt(V) = C - sqrt((g*pi*r^2)/2)*t *(r/R) ??? factor 1/sqrt(2) does not look right

where C = c/2 is just another way of writing the constant.

V(t) = [C - sqrt((g*pi*r^2)/2)*t *...]^2

If, at time t = 0, V(0) = Vo, then: define V0 in therms of h(0)

Vo = C^2

C = sqrt(Vo)

so:

V(t) = [sqrt(Vo) - sqrt((g*pi*r^2)/2)*t *...]^2

After this i am supposed to form a table of values in excel but when i do i do not seem to get the right values. I put time in the a column and this equation in the b column and A1 or time for t and 9.8 for g. any suggestions or help would be greatly appreciated

Cylinder radius R and hole radius r must have different symbols - they are not the same. From the statement above where I said NO, everything is off by a factor of r/R, and from ??? onward another factor of 1/sqrt(2) [Check that I did it right, please!]

Does that help get better values in Excel?