Geology 351; Final Exam Review
Example Questions
Some calculus questions
9.8 Consider example problem 9.8 in the text (page 171).
The basic relationship is
3
20
zkW
Q
km
=
. This tells you that heat is being generated at a
certain rate per cubic kilometer. This rate varies with distance above the base of the crust
(z). The crust in this example is assumed to be 30km thick. There are some typical
questions that can be asked about Q. Q gives us an amount of power per unit volume. So
we might ask what the amount of heat energy per unit time would be for a certain
specified volume. Since the heat generation rate varies with z, this is not a simple Q times
volume answer. One way is to add up a lot of little volumes. Each little volume generates
a different amount of heat since it lies at a different depth. So the total heat generated
over a certain interval would be
1
n
i
i
qQ
=
z
=
Δ
∑
. Now, a question you could be asked is how
to express this sum as an integral? The integral is identical to this sum in a sense, except
that we let our little volume element shrink to 0.
i) What would that integral be?
ii) Express that integral in definite form where the integration takes place from a range of
distances z
1
to z
2
above the base of the crust.
iii) Then you would be expected to evaluate that integral. If we wanted to calculate the
total heat generated by a column of crust extending from the base of the crust to the
surface  what would that be?
9.9 Since the Earth is an oblate spheroid, accurate estimate of its volume should take this
into account. The Earth is not a perfect sphere but is flattened at the poles by about 21
km. The distance from the Earth’s axis of rotation to the surface at varying positions from
pole to pole, varies from 0 to r
e
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 Spring '09
 Radioactive Decay, Geology, Orders of magnitude, North Pole, South Pole

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