One rod is made from lead and another from quartz. the rods are heated and experience the same change in temperature. the change in length of each rod is the same. if the initial length of the lead rod is 0.10 m, what is the initial length of the quartz rod?

I think there is a mistake in the subject area. This has nothing to do with geography, but with physics. This topic is about the linear expansion of conductive materials when heated to a higher temperature. The equation for linear expansion is

ΔL = L0*α*ΔT, where
ΔL is the change in length when heated
L0 is the original length of the rod
α is the coefficient of linear expansion; this is an empirical data for a specific kind of material
ΔT is the change in temperature

Since ΔL is same for lead and quartz, we can equate them.

L0,lead*α,lead*ΔT = L0,quartz*α,quartz*ΔT

The α for lead is 29×10^6 °C/m while the α for quartz is 14×10^6 °C/m. Since ΔT are the same for both materials, ΔT will be cancelled out. Therefore, the equation for L0,quartz is:

L0,quartz = (L0,lead*α,lead)/α,quartz
L0,quartz = (0.10 m*29×10^6 °C/m)/14×10^6 °C/m
L0,quartz = 0.207 m

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