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Rates in Geology

Practice problems

Practice calculating rates (and rearranging the rate equation) below using the "rules" that you have just learned. Answers are provided (but try doing them on your own before peeking!).

Calculating rates


Problem 1:
You wake up at 6 am (EARLY!) and the temperature is 55°F. By the time you head off to your picnic lunch at noon, the temperature has risen to 91°F. What is the rate of temperature change in °F per hour?

  1. Determine which changing variable is ΔX and which is Δt.
    In the above, you need to calculate a change in temperature, so this will be ΔX and the amount of time that has elapsed, Δt.
  2. Calculate ΔX and Δt to determine the change in the variable(s).

  3. Lets begin by calculating the change in temperature.
    ΔX=91°F - 55°F = 36°F
    Next we need to calculate the amount of time that elapsed between getting up and heading off to your picnic. In other words, how many hours between 6 am and noon?
    Δt=12 - 6 = 6 hours
  4. Calculate the rate using ΔX and Δt.
  5. The rate is ΔX/Δt, so
    temperature change rate
    Dividing, the rate is 6°F per hour.
  6. Check to see what the units on your final number should be before continuing.
  7. In this case, we are asked to calculate a rate in °F per hour. Do we have units of °F and hours? Yes!
  8. Evaluate your answer. Does your answer seem reasonable?
    6°F per hour is a reasonable number since over a couple of hours the temperature would change about a dozen degrees.


Problem 2.The Hawaiian hot spot sits below the Pacific plate. As the plate moves over the hot spot, a chain of volcanoes is formed. The Waianae volcano on Oahu is 3.7 million years old and about 375 km from the current location of the Hawaiian hot spot. Assuming that the hot spot is in a fixed location, how fast (at what rate) is the Pacific plate moving?

  1. Determine which changing variable is ΔX and which is Δt.
    In the above, you need to calculate a change in location which is distance, so this will be ΔX and the amount of time that has elapsed, Δt.
  2. Calculate ΔX and Δt to determine the change in the variable(s).
  3. In this case ΔX and Δt are given - no calculations are necessary. ΔX is given as the distance (375 km) and Δt the time (3.7 million years).
  4. Calculate the rate using ΔX and Δt.
  5. The rate equation is
    r=x/t
    substituting in,

    R=kmpermy
    so
    R= 101.4 km/million years

    (This is about 10.1 cm/yr)
  6. Check to see what the units on your final number should be before continuing.
  7. In this case, no specific units are requested, so no need to worry about this!
  8. Evaluate your answer. Does your answer seem reasonable?
    Plates typically move 10-150 km/my, so this seems reasonable.


Problem 3:
The Hawaiian hot spot has produced about 775,000,000 km3 of magma in the past 70 million years. What is the average rate of magma production per year?

Remember to follow the steps for calculating a rate:

  1. Determine which changing variable is ΔX and which is Δt.
    In this case the two values are volume for ΔX and, as is typical, time for Δt.
  2. Calculate ΔX and Δt to determine the change in the variable(s).

  3. In this case since the volume has gone from 0 km3 to 775,000,000 km3, so ΔX = 775,000,000 km3. Δt = 70,000,000 years because that is the amount of time that has elapsed.
  4. Calculate the rate
  5. The equation for rate is:
    r=x/t
    Substituting in for ΔX and Δt,
    hotspot rate
    so the average rate of magma production is 11 km3/yr.
  6. Check to see what the units on your final number should be before continuing.
  7. The question asks for the average rate of magma production per year, so the final units should be km3/yr, and it is!
  8. Evaluate your answer
  9. This is a bit difficult in this problem, since it is hard to know what a reasonable amount of magma production in a year is. However, if you had gotten a very small number (like .001 km3/yr), you might realize that there is no way that a giant set of volcanic islands could be made with that amount of magma.

    Another way to evaluate your answer is to make a quick estimation. If the volcano erupts 10 km3 per year (close to the answer you got) for 70 million years, that would be get 700 million km3 of magma. This is about what the question says is the amount, so it seems this is a reasonable answer.


Problem 4: Rivers often form sinuous paths as they flow downstream. The river bends are called meanders and move over time as the river erodes its banks. In 2010, you purchased a house that was 150 meters from the outside of a meander of the Rio Grande river. Looking at maps from 1955, you find that the meander has moved 230 meters toward you. How fast is the meander migrating?

  1. Determine which changing variable is ΔX and which is Δt.
  2. In this case the distance is ΔX and as always the time elapsed is Δt.
  3. Calculate ΔX and Δt to determine the change in the variable(s).

  4. The distance (ΔX) is clear - 230 meters - but the time may not be so clear. The river moved this distance between 1955 and 2010, so the time (Δt) is the difference between these, found by subtraction:

    Δt= 2010-1955 = 55 years.
    So the river moved 230 meters in 55 years.
  5. Calculate the rate
  6. The equation for rate is r=x/t
    Substituting in for ΔX and Δt,
    R = 230 m/55 yr,
    So, the rate is 4.18 m/yr
  7. Check to see what the units on your final number should be before continuing.
  8. In this case, no specific units are requested, so no need to worry about this!
  9. Evaluate your answer
  10. 4.18 m is about 13 feet. While this is a lot of erosion, it is certainly reasonable.

Determining a rate from a graph


Problem 5:Examine the graph of the age and distance of the New England Seamounts. This chain of seamounts are thought to be created by a hotspot that underlies the oceanic plate that they sit on. As the plate moves, it carries the seamounts with it. What is the rate of movement that they show? (You can click on the graph for a bigger version)


To solve this problem, first pick any two points on the line and then determine the slope of the line, which is the rate.

New England Seamounts with markedpoints Age and location of the New England Seamounts with two points picked in blue
You can use any two points that you would like. The farther apart they are, however, the less likely you are to make mistakes that will matter in the end. Two points are marked in blue on the graph to the right. The upper right point is at about 104 million years and 1030 km and the lower left point is at 81 million years and 0 km

New England Seamounts with plotted points Red lines to the horizontal and vertical axes show how the coordinates of the points were determined to be (104 million years, 1030 km) and (81 million years, 0 km).



Slope is rise divided by run. The rise is the difference between the vertical values (the distances), and the run is the difference between the horizontal values (the distances).
The rise then is

104 my- 81 my = 23 my
(my is million years)
The run will be
1030 km - 0 km = 1030 km
Dividing the two, (rise divided by run)
1030 km/23 my

= 44.8 km/my



The slope is equal to the rate as long as the horizontal axis is time. So in this case the rate is the slope!
The plates velocity is 44.8 km/my.
Age and location of the New England Seamounts
Figure modified from Duncan 1984. J. Geophys. Res. 89 (B12):9980-9990 and from http://oceanexplorer.noaa.gov/explorations/03mountains/background/geology/geology.html


I think I've mastered these rate problems! Let me try the quiz!
(If this is not how you feel see the links below for more practice!)

Still need more practice?


There are many web sites and books that walk you through the rates problems, although most will be distance, velocity and time problems. However, the mathematics is identical.


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