help with homework

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March 9th 2006, 02:05 AM
charps43

help with homework

I'm trying to finish a math project that's due in a few hours. I really stuck on a key aspect of it. It is called "Bowl Lake". Its called that because it geometrically can be viewed as having a bottom contour that resembles a portion of a sphere. The lake is 35 ft deep at its deepest point(the center) and 1/4 mile across(1320 feet).

My big problem is finding the volume of the lake. I can't seem to find the equation for the integral or figure out the dimensions of the sphere that this lake is a part of. Was hoping someone could give me a hint or something. Thanks alot.

Brett
March 9th 2006, 02:42 AM
CaptainBlack

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Quote:

Originally Posted by charps43

I'm trying to finish a math project that's due in a few hours. I really stuck on a key aspect of it. It is called "Bowl Lake". Its called that because it geometrically can be viewed as having a bottom contour that resembles a portion of a sphere. The lake is 35 ft deep at its deepest point(the center) and 1/4 mile across(1320 feet).

My big problem is finding the volume of the lake. I can't seem to find the equation for the integral or figure out the dimensions of the sphere that this lake is a part of. Was hoping someone could give me a hint or something. Thanks alot.

Brett

The first thing to do is determin the radius of the sphere that the lake is
a part of. See the attached diagram for details or the crossection of lake
and sphere.

The radius of the sphere can be found from the intersecting cord theorem.
In this case it tells us that:

$<br /> (2R-35)25=\left(\frac{1320}{2}\right)^2<br />$

Having forund $R$ the volume of the lake can be found using the
formula for the volume of a spherical cap:

$<br /> V=\frac{1}{3}\pi h^2(3R-h)<br />$ ,

where $R$ is the radius of the sphere and $h$ is the height of the cap (in this case
the depth of the lake).

RonL
March 9th 2006, 02:46 AM
CaptainBlack

Please do not post the same problem multiple times. The other copy of
this question has been deleted.

RonL
March 9th 2006, 02:46 AM
charps43

you're the best. Thank you very much!

brett
March 9th 2006, 03:14 AM
ticbol

Quote:

Originally Posted by charps43

I'm trying to finish a math project that's due in a few hours. I really stuck on a key aspect of it. It is called "Bowl Lake". Its called that because it geometrically can be viewed as having a bottom contour that resembles a portion of a sphere. The lake is 35 ft deep at its deepest point(the center) and 1/4 mile across(1320 feet).

My big problem is finding the volume of the lake. I can't seem to find the equation for the integral or figure out the dimensions of the sphere that this lake is a part of. Was hoping someone could give me a hint or something. Thanks alot.

Brett

Here is one way.

You need hint/hints only? Not my type, but I am busy reading something on a separate website, so here are the hints:

1.) If you do not want to do Calculus, you can find the volume of your lake by using the formula for the volume of a spherical cap,
V = pi(h/6)(3r^2 +h^2) --------***
where
h = heigth or depth of the cap, which is 35ft in your lake.
r = radius of the cap's lid, which is 1320/2 = 660ft in your lake.

So your lake's volume is
V = pi(35/6)(3*660^2 +35^2) = 23,970,810 cu.ft. ----answer.

2.) If you are really into Calculus, then find the radius R of the sphere where the cap (your lake) is a part of, so that you can find the equation of the big circle of the sphere, etc...(maybe using horizontal circular discs as your volume differentials, or dV, then you integrate from y = -R to y = -(R-35), etc...)

You can find R by using a secant of the big circle of the sphere,
R^2 = (660)^2 +(R-35)^2 ----------***