Volume of a frustrum as function of height

[mathero]

Hi
I am following the book Calculus made easy by Silvanus P. Thompson and I came across this exercise:



I understand everything but the bit where it uses the mean level = 12 feet.
I would have thought that the difference in level is 4 (14-10).
Why does the author use the mean?

Thanks.

 

[Deleted member 4993]

Hi
I am following the book Calculus made easy by Silvanus P. Thompson and I came across this exercise:

View attachment 31640

I understand everything but the bit where it uses the mean level = 12 feet.
I would have thought that the difference in level is 4 (14-10).
Why does the author use the mean?

Thanks.

Let's attack the problem slightly differently. Do you understand that the volume of water, in the frustum at a depth 'd' is V = 40000*d + 400*d^2 +4*d^3/3

When d = 14 let V = V₁₄ = 40000*14 + 400*14^2 +4*14^3/3 = 642058.6667 cft

When d = 10 let V = V₁₀ = 40000*10 + 400*10^2 +4*10^3/3 = 441333.3333 cft

So cft of water lost due to change in height from 14' to 10' = 200725.3333 cft = 1254533.333 gallons (in 24 hrs.)

Gallons per hour = 52272.22222

Thus we get the same answer (except for rounding error - I did my calculations in Excel and pasted the answers here without rounding). The "averaging" works because the walls of the frustum are flat. If those were part of an ellipsoid or paraboloid - the answers will be different. As we can visualize, the rate of water loss is NOT constant - but changing continuously (just like real speed of a car - we call that good enough for government work).

Good question......

 

[mathero]

Thanks very much @Subhotosh Khan. Great explanation!

I was scratching my head for a good day. Clearly I forgot about something very fundamental "the water loss is not constant" and the flat walls hence the mean can be used.
Thanks for explaining another way of getting to the same result. That indeed reinforces what has been done with the mean.

 

[lex]

The two answers are very close, but the difference isn't due to rounding error. The function [imath]\tfrac{dV}{dh}[/imath] is practically linear between 10 and 14 but not quite. The mean value of the function over the interval from 10 to 14 is [imath]50181\tfrac{1}{3}[/imath] (as can be seen by taking [imath]642058\tfrac{2}{3}-441333\tfrac{1}{3}[/imath] and dividing by 4 ; Subhotosh Khan's work above). The mean value of the function over the interval 10 to 14 is not the value of the function at the mean of 10 to 14. The value at 12 is 50176, but the mean value (achieved about 12.06) is [imath]50181\tfrac{1}{3}[/imath].
This explains the difference, but obviously as said above 'good enough for government work'!