ANSWER KEY for Question #9 of Lab #3 (Materials of the Sea Floor)
Online Oceanography 101 (Gwyneth Jones) - Bellevue Community College
9) Red clay on the deep-sea floor accumulates at an average rate of about 1 millimeter per 1,000 years. Biogenous oozes often accumulate at rates that are ten times faster than red clay.
a) How much time would be required to deposit 5 centimeters of red clay?
b) How much total accumulation time would be represented by a core 10 meters in length if it contained 5 meters of red clay and 5 meters of ooze?
Two versions of the answer key -- Take your pick:
Short version (Choose the short version if you consider yourself good at math and think you just made a simple math mistake when you tried this on your own) Long version (Choose the long version if you're rusty on math or don't like math, and want to see how I would think through this problem, step by step)
9a)
5 cm = 50 mm
50 mm x (1,000 yr / 1 mm) = 50,000 years
9b)
Red clay:
5 m of red clay = 5,000 mm x (1,000 yr / 1 mm) = 5,000,000 years
Biogenous ooze:
Rate is 10x the red clay rate = 1 mm / 100 yr = 1 cm / 1,000 yr
5 m of biogenous ooze = 5,000 mm x (100 yr / 1 mm) = 500,000 years
Total = 5,000,000 yr + 500,000 yr = 5,500,000 years
STEP 1 - Always start by figuring out what the question is asking for. (What kind of answer? -- A time? A speed? A distance? Something else?)
STEP 2 - Then list the information that the question provides. (What's given?)
STEP 3 - Next, figure out what additional information you might need to answer the question. (What else do you need to know? Equations, unit conversions, etc.)
STEP 4 - Then -- BEFORE doing any math! -- do a "very rough estimate" of what the answer will be. (What will the answer look like? -- Many years? A slow rate? A huge distance? A tiny distance? Something else?)
STEP 5 - Next, set up the calculation, showing all units. (Write it all out neatly, so you'll remember later what you did.)
STEP 6 - Then, plug in the numbers. (This is where some students try to start, but it really and truly saves time to do the set-up steps methodically!)
STEP 7 - Finally, check to be sure your answer makes sense. (Compare your STEP 4 estimate with your STEP 7 calculation. "Did I say that plates move millions of kilometers a year, when I know that they move just centimeters a year? If so, where did I go astray?")
STEP 8 - Fix any mistakes that you might have made.
STEP 9 - It generally helps to circle/box your answer, but it's not mandatory.
Now let's do Q.#9a
STEP 1 - Always start by figuring out what the question is asking for.
We're looking for an amount of time in this question - How long it takes to accumulate 5 cm of red clay on the deep-ocean floor.
STEP 2 - Then list the information that the question provides.
Accumulation rate for red clay = 1 mm per 1,000 yrs
Accumulating 5 cm of red clay
STEP 3 - Next, figure out what additional information you might need to answer the question.
We're given a rate and a distance, and asked to find a time.
Equation: Distance (thickness of sediment) = Rate (accumulation rate) x Time (years to accumulate)
Unit conversion: 1,000 mm = 1 m
Unit conversion: 100 cm = 1 m (If you don't think of this now, that's fine -- it'll probably become apparent in Step 5. Just write it up with the other Step 3 information, so it's all in one place.)
STEP 4 - Then -- BEFORE doing any math! -- do a "very rough estimate" of what the answer will be.
That's a very slow accumulation rate (a millimeter is a only a tiny fraction of an inch; it's the smallest "tick mark" on the metric side of a ruler). We're asked about accumulating 5 cm of sediment (so, a couple inches). So, the answer will be something on the order of "many years", rather than "a few minutes".
STEP 5 - Next, set up the calculation, showing all units.
Distance = Rate x Time
Given D and R, looking for T, so rearrange (divide both sides by R to get T alone):
Distance / Rate = (Rate x Time) / Rate
Distance / Rate = Time
Time = (5 cm) / (1 mm / 1,000 yrs) -- (This would look a lot prettier on paper, where you can put the top part of the fraction above a horizontal line, rather than after a slash.)
STEP 6 - Then, plug in the numbers.
Uh oh, what do we do about those units? - There are several ways to approach this... (a) You can convert the 5 cm into mm, or (b) You can convert the 1 mm into cm. Either way works.
(a) 5 cm x (1 m / 100 cm) x (1,000 mm / 1 m) = 50 mm (So, 5 cm = 50 mm)
or
(b) 1 mm x (1 m / 1,000 mm) x (100 cm / 1 m) = 0.1 cm (So, 1 mm = 0.1 cm)
If you did (a) -- Time = (50 mm) / (1 mm / 1,000 yrs) = 50,000 years
or
Time = (50 mm) x (1,000 yrs / 1 mm) = 50,000 years
or
If you did (a) -- Time = (5 cm) / (0.1 cm / 1,000 yrs) = 50,000 years
or
Time = (5 cm) x (1,000 yrs / 0.1 cm) = 50,000 years
STEP 7 - Finally, check to be sure your answer makes sense.
We estimated "many years" in STEP 4, so yes. The answer is in years, and it's a lot of them.
STEP 8 - Fix any mistakes that you might have made.
No mistakes this time! But really, it's easy to make some mistakes along the way, like dividing instead of multiplying (hence my recurring advice to write everything out, using "--------------" instead of " / " to show fractions, and check to make sure that units really are cancelling properly).
STEP 9 - It generally helps to circle/box your answer, but it's not mandatory.
It takes 50,000 years to accumulate 5 cm of red clay on the deep-sea floor.
Okay, everyone have a headache now? Congratulations for reading through to the end of that explanation! It's easier to explain math in person and on paper, at least for me!
And let's do Q.#9b
STEP 1 - Always start by figuring out what the question is asking for.
We're looking for an amount of time again - How long it takes to accumulate a total of 10 meters of sediment, half of which is red clay (like in Q.#9a) and half of which is biogenous ooze (different accumulation rate).
STEP 2 - Then list the information that the question provides.
Accumulation rate for red clay = 1 mm per 1,000 yrs
Accumulating 5 m of red clay
Accumulation rate for biogenous ooze = Ten times faster than the red clay rate
Accumulating 5 m of biogenous ooze
STEP 3 - Next, figure out what additional information you might need to answer the question.
We're given rates and distances, and asked to find a time. Looks like this is a two-part calculation -- How long to accumulate the red clay and how long to accumulate the biogenous ooze. Luckily, we calculated the same sort of thing for red clay in Q.#9a, so some of the set-up should be similar.
Equation: Distance (thickness of sediment) = Rate (accumulation rate) x Time (years to accumulate)
Unit conversion: 1,000 mm = 1 m
STEP 4 - Then -- BEFORE doing any math! -- do a "very rough estimate" of what the answer will be.
Definitely going to be "many, many years", because it took 50,000 years to accumulate just 5 centimeters of red clay (Q.#9a), and here we're looking at 5 meters of red clay (100 times more of it!) plus 5 meters of biogenous ooze. (Our answer should end up being more than 100x the 50,000 years we got in Q.#9a -- because a meter is one hundred times bigger than a centimeter, and because we have the biogenous ooze to account for, as well.)
STEP 5 - Next, set up the calculation, showing all units.
Red clay: Time = (5 m) / (1 mm / 1,000 yrs)
Biogenous ooze: Time = (5 m) / (uh-oh, what's the rate??? - yeah, maybe we should have figured this out sooner, but hey, at least we thought of it now!)
We're told that the biogenous ooze rate is TEN TIMES FASTER than the red clay rate. So, the accumulation times for equal amounts of the two types of sediments will be quite different. Biogenous oozes are speedier, essentially. It will take ONE-TENTH the time to accumulate an equal amount of biogenous ooze. Or you can say, it will take 10x LONGER to accumulate an equal amount of red clay. Or, in the same amount of time, TEN TIMES as much biogenous ooze will accumulate.
So, biogenous ooze rate = 10 x red clay rate
Biogenous ooze rate = 10 x (1 mm / 1,000 yrs) = 10 mm / 1,000 yrs = 1 mm / 100 yrs
Biogenous ooze: Time = (5 m) / (1 mm / 100 yrs)
STEP 6 - Then, plug in the numbers.
Red clay: Time = (5 m) / (1 mm / 1,000 yrs)
5 m x (1,000 mm / 1 m) = 5,000 mm (So, 5 m = 5,000 mm)
Time = (5,000 mm) x (1,000 yrs / 1 mm) = 5,000,000 years
So, it takes 5,000,000 years to accumulate 5 meters of red clay
But that only accounts for half of the total 10 meters of sediment in the question...
Biogenous ooze: Time = (5 m) / (1 mm / 100 yrs)
5 m x (1,000 mm / 1 m) = 5,000 mm
Time = (5,000 mm) x (100 yrs / 1 mm) = 500,000 years
So, it takes 500,000 years to accumulate 5 meters of biogenous ooze
STEP 7 - Finally, check to be sure your answer makes sense.
We estimated "many, many years" in STEP 4, so yes.
We knew that "It will take ONE-TENTH the time to accumulate an equal amount of biogenous ooze", and 500,000 years is indeed 1/10 of 5,000,000 years, so that's good, too.
STEP 8 - Fix any mistakes that you might have made.
We haven't written down the FINAL ANSWER yet (we "should have" done that in STEP 6).
Total = Red clay + Biogenous ooze = 5,000,000 years + 500,000 years = 5,500,000 years
STEP 9 - It generally helps to circle/box your answer, but it's not mandatory.
It takes 5,500,000 years to accumulate a core of sediment on the deep-sea floor consisting of 5 meters of red clay and 5 meters of biogenous ooze.