Simple Rate Problems in Geology

One of the most-common uses of mathematics in introductory geology courses is in looking at rates.  Here are some basic examples:

1. Sediment is accumulating in a deep ocean basin at an average rate of 2 mm/1,000 years.  How much sediment will accumulate in a million years?

2. Assume an average erosional rate of rock to be 0.1 mm/yr.  How many years will it take to erode away a 3,500 m high mountain peak?

3. A basaltic lava flow cools from 1,200° C to 20° C in 36 hours.  What is the average rate of cooling in °C/min?

4. A volcanic island on the Pacific Plate has moved 200 km away from a hot spot.  The oldest volcanics on the island are 5 million years old.  What is the average rate of plate movement in cm/yr?

5. If sea level rises at at average rate of 2 mm/yr, how long before an area 25 meters above sea level will be inundated?

Now, in some ways, these are unrealistic problems because they assume average linear rates of change.  In reality, geologic processes are generally not linear.  For example, if we look at the rate at which sediment accumulates in the Mississippi Delta area, we'll see that it's highly variable based upon the flow of the river.  During times of drought, the rate slows way down and during flood stages, the rate may be incredibly high.  Nevertheless, it is still sometimes helpful to look at average rates of change because they give an idea of how long things take to happen (how long, for example, is needed to completely erode away a mountain belt?).

Anyway, these problems are common in introductory geology lab manuals and I'm always surprised that some college-level (supposedly) students will struggle with these middle-school-level math problems.  So, a few notes about how to approach such things...

First, recognize that these problems are all of the same form - they involve a rate (R) which is a change in distance (D) units over (divided by) time (T) units.  In other words:

   R = D / T

Using basic algebra, we can rearrange this to solve for either time (T) or distance (D):

   T = D / R
   D = R T

Next, and some students have trouble with this, we need to read the word problem and identify the three things - R, D, and T.  One of these we'll be solving for and the other two are given.  For example, in the five examples given above:

   1. R = (2 mm/1,000 yr); D = ?; T = 1,000,000 yr
   2. R = (0.1 mm/yr); D = 3,500 m; T = ?
   3. R = ?; D = (1,200° C - 20° C); T = 36 hr
   4. R = ?; D = 200 km; T = 5,000,000 yr
   5. R = (2 mm/yr); D = 25 m; T = ?

One that might have been a little tricky is recognizing in number 3 that the D value is the change in temperature (think of it as the distance between the starting and ending temperature).

Now we know which equation to use depending on whether we're solving for R, D, or T.

Another place where people get into trouble.  They'll plug in the numbers without thinking about the units.  For example, in solving example 5 above, they'll write:

   T = D / R = 25 / 2 = 12.5

Of course, I'll respond with 12.5 what?  Years?  No.  You divided meters by millimeters per year so your answer is 12.5 meter year per millimeter.  What the hell is that?

You MUST pay attention to units when solving these problems.  If you have one variable in meters and another in millimeters, you have to either convert to either one or the other for consistency.  You also need to pay attention to what units your answer should be in and convert accordingly.

Also, if you're not told what units your answer should be in, use common sense.  Suppose you're solving for time and the answer comes out to 4,320 minutes.  That's not useful to most people but converting it to 72 hours or 3 days is much better.

Let's solve the five example problems:

1. Sediment is accumulating in a deep ocean basin at an average rate of 2 mm/1,000 years.  How much sediment will accumulate in a million years?

   How much sediment is D = R T
   D = (2 mm/1,000 yr) (1,000,000 yr)  [Should be able to solve this in your head!]
   D = 2,000 mm
   Since units aren't specified for the answer, I'd covert answer to meters
   2 meters of sediment will accumulate

2. Assume an average erosional rate of rock to be 0.1 mm/yr.  How many years will it take to erode away a 3,500 m high mountain peak?

   How many years is T = D / R
   Problem! R is in millimeters and D is in meters.  Need to convert
   3,500 m (1,000 mm/m) = 3,500,000 mm
   T = 3,500,000 mm / 0.1 mm/yr = 35,000,000 yr
   35 million years to erode away the peak

3. A basaltic lava flow cools from 1,200° C to 20° C in 36 hours.  What is the average rate of cooling in °C/min?

   Rate problem is R = D / T
   D is difference in temperatures so 1,200° C - 20° C = 1,180° C
   We want our answer in minutes so 36 hr (60 min/hr) = 2,160 min
   R =  1,180° C / 2,160 min = 0.5462962962962962962962962962963° C/min
   Why the long number?  That's what my calculate said.  It's ridiculous, we need to round.
    Since temperature and time are only given as whole numbers, round to one digit.
    R = 0.5° C/min
   Half a degree Celsius per minute cooling

4. A volcanic island on the Pacific Plate has moved 200 km away from a hot spot.  The oldest volcanics on the island are 5 million years old.  What is the average rate of plate movement in cm/yr?

   This again asks for a rate so R = D / T
   The answer needs to be in cm/yr so D needs conversion from km to cm
   200 km (100,000 cm/km) = 20,000,000 cm
   R = 20,000,000 cm / 5,000,000 yr = 4 cm/yr
   The average rate of plate movement is 4 cm/yr

5. If sea level rises at at average rate of 2 mm/yr, how long before an area 25 meters above sea level will be inundated?

   A time problem so T = D / R
   Need to get distance and rate in the same units so let's convert m to mm
   25 m (1,000 mm/m) = 25,000 mm
   T = 25,000 mm / 2 mm/yr =12,500 yr
   It will take 12,500 years for sea level to flood this area

These are all reasonable problems and kind of neat to think about.  Sediment does accumulate in ocean basins (and other environments).  Rocks and even entire mountain ranges do erode away.  Igneous rocks cool from their molten state.  Volcanic islands, like Hawaii, do move off of hot spots with the motion of the Pacific Plate.  Sea level has been rising since the last ice age (and will continue to do so at a faster rate with anthropogenic warming of the climate).

It's interesting to think about how long those things take to happen.  While eroding a mountain in 35 million years seems like a long time, remember that in the 4.5 billion year history of the Earth, you could have eroded away well over 100 consecutive mountain ranges!  While a plate movement of 4 cm/yr seems slow, a plate could move 2600 km (the distance across the Atlantic Ocean) since the time of the extinction of the dinosaurs 65 million years ago (and, geologically, that wasn't very long ago!).  The 25 meter rise in sea level could have taken place since the end of the last ice age.

In other words, even though some of these rates seem slow, geologists have plenty of time to play with and even small rates add up to big changes over geological time spans.