### Smallish Numbers

So I was sitting on the beach down at Tropical Island Paradise last week with the family, watching the pitiful little six-inch tropical tide, and thinking about our more he-man sized tides back here in the Smallish State. It occurred to me that, with the longest coastline in the Lower 48, and the highest tides, The S. State must may have the most variable surface area of all the states. Which observation of course leads to the question: how much bigger is the Smallish State at low tide, compared to high tide? Does it add appreciably to our area when the tide is down?

There are some challenging variables to guess at in coming up with an estimated answer. First: how much coastline does the S.S. actually have? According to the state website, “over 5,000 miles”, if you include all the islands, which of course we will. Next: what is the average tidal rise & fall of the entire coastline? Hard to say exactly, but the tide does vary somewhat linearly along the coast—from an average of about 9ft at the southwest border, to an average of about 19ft at the northeast tip. So as a rough average of averages, let’s call it 14 feet. Now the haziest factor: in order to decide what width of beach/rock/mud/lobster is exposed when the tide goes down 14 vertical feet, you need first to guess at the average angle that the shore slopes off into the ocean. In some places, it’s virtually vertical (on Sandra Lee, at times, we’ve been within 30ft of shore in 150ft of water.) In other places, like the southern beaches, it looks to be no more than 5 or 10 degrees of slope. In most places, though, I’d say you can cautiously wade out into the water up to your chin, without precipitously falling down an underwater ski-slope—which suggests to me that the average might be no more than 30 degrees. So, let’s call it that (anyone with better information, please comment.)

Now then. Trigonometry provides that the hypotenuse of a 30 degree right triangle will be twice as long as the short side opposite the 30 degree angle (sin30=0.5=opposite/hypotenuse.) So a 14ft vertical drop on a 30 deg sloping shore will unveil 28 feet of land. (Does this seem about right? At low tide, the distance from the water’s edge to the line of old driftwood, about five body lengths? Yes, I think so.) 28 feet is (28/5280=) .0053 miles. So along every mile of shore, (1mi x .0053mi=) .0053 square miles of land dries out each low tide. And there being 5,000 miles of coast, this totals (5000 x .0053=) 26.5 square miles uncovered at each low tide. Or 16,960 acres, which sounds a bit more impressive.

How much is this in relation to the Smallish State’s overall area? Pretty negligible, it turns out—the area of the state is over 3,000 square miles, so adding 26 doesn’t change much. But, the Smallish City, true to its name, occupies only 21 square miles. And we’d hate for those particular square miles to go under water twice a day.

There are some challenging variables to guess at in coming up with an estimated answer. First: how much coastline does the S.S. actually have? According to the state website, “over 5,000 miles”, if you include all the islands, which of course we will. Next: what is the average tidal rise & fall of the entire coastline? Hard to say exactly, but the tide does vary somewhat linearly along the coast—from an average of about 9ft at the southwest border, to an average of about 19ft at the northeast tip. So as a rough average of averages, let’s call it 14 feet. Now the haziest factor: in order to decide what width of beach/rock/mud/lobster is exposed when the tide goes down 14 vertical feet, you need first to guess at the average angle that the shore slopes off into the ocean. In some places, it’s virtually vertical (on Sandra Lee, at times, we’ve been within 30ft of shore in 150ft of water.) In other places, like the southern beaches, it looks to be no more than 5 or 10 degrees of slope. In most places, though, I’d say you can cautiously wade out into the water up to your chin, without precipitously falling down an underwater ski-slope—which suggests to me that the average might be no more than 30 degrees. So, let’s call it that (anyone with better information, please comment.)

Now then. Trigonometry provides that the hypotenuse of a 30 degree right triangle will be twice as long as the short side opposite the 30 degree angle (sin30=0.5=opposite/hypotenuse.) So a 14ft vertical drop on a 30 deg sloping shore will unveil 28 feet of land. (Does this seem about right? At low tide, the distance from the water’s edge to the line of old driftwood, about five body lengths? Yes, I think so.) 28 feet is (28/5280=) .0053 miles. So along every mile of shore, (1mi x .0053mi=) .0053 square miles of land dries out each low tide. And there being 5,000 miles of coast, this totals (5000 x .0053=) 26.5 square miles uncovered at each low tide. Or 16,960 acres, which sounds a bit more impressive.

How much is this in relation to the Smallish State’s overall area? Pretty negligible, it turns out—the area of the state is over 3,000 square miles, so adding 26 doesn’t change much. But, the Smallish City, true to its name, occupies only 21 square miles. And we’d hate for those particular square miles to go under water twice a day.

## 5 Comments:

whew, math posts make my head hurt! i thought you were suppose to be on VACATION last week, not crunching numbers.

What gt. said! I skimmed once the numbers got too intense so my head would not explode. Nonetheless, my right-brained self thanks you for the insight into the inner workings of a left-brained clinician-scientist. Apparently it's scary in there...

All of that on the beach?? I was sitting next to you also making important considerations; rum punch or Corona or pina colada? Nap or read? 15 sunscreen or 30?

you are wonderfully and beautifully obtuse. the fact that you remember equations from trig...well, let's just say that it adds exponentially to your uniqueness :)

But, the Smallish City, true to its name, occupies only 21 square miles. And we’d hate for those particular square miles to go under water twice a day.Give it a decade or three, and half of it will be under water all the time.

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