How long will it take for 750 mg of a sample of radium-225, which has a half-life of about 15 days, to decay to 68 mg?
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Answer:
Step-by-step explanation:
Given
Required
Determine an equivalent expression
In trigonometry:
In
Substitute for
in
Hence, the equivalent expression is:
Because I've gone ahead with trying to parameterize directly and learned the hard way that the resulting integral is large and annoying to work with, I'll propose a less direct approach.
Rather than compute the surface integral over straight away, let's close off the hemisphere with the disk
of radius 9 centered at the origin and coincident with the plane
. Then by the divergence theorem, since the region
is closed, we have
where is the interior of
.
has divergence
so the flux over the closed region is
The total flux over the closed surface is equal to the flux over its component surfaces, so we have
Parameterize by
with and
. Take the normal vector to
to be
Then the flux of across
is