Economics 101, a Novel, ch_39 -- Trig in the Morning, and Health and Safety

(The story starts here: http://free-is-not-free.blogspot.jp/2016/03/economics-101-novel-ch00.html.

The previous chapter is here: http://free-is-not-free.blogspot.com/2016/07/economics-101-novel-ch38-changing.html.)

[JMR201610140712: I'm definitely losing the thread by the end of this chapter. I'm not sure how far back I have to go to get the thread back.]

Just how useless is trigonometry, really?



"Bobbie? Are you awake?"

"Barely. Are we going back up to the ridge today to look at that island again?"

"We probably should. It's way out there, you know."

"How far, do you think?" Bobbie asked.

"I think there's an easy formula I've forgotten, but let's work it out. Where's a stick for some drawing?" Karel got up and started looking for a stick.
Bobbie came around the tent with one in her hand and they sat down side-by-side on the ground. Bobbie drew a circle in the dirt, with a diameter line.

"Okay, here's the earth, and the diameter at the horizon where we thought we saw the island." And she drew another line. "Here's the diameter plus the height of the ridge, at the mountain top."

"Right, if what we saw was not a mirage, and not accounting for the refraction of the atmosphere. Basic stuff. But we need to pray, I think."

"I'll say it."

They both bowed their heads and Bobbie said, "Heavenly Father, we thought we saw another island in the ocean, west of here. But Wycliffe said this island is still uncharted, and we've been busy. Anyway, we'd sort-of forgotten it until Karel's strange dream last night. We need to know how far it might be. And whether we should try to visit it. Please guide us to get this calculation right, so we won't put ourselves in too much danger or ignore possibilities that might help us." And she closed the prayer and they both said "Amen".

"Okay," she continued, "here's the tangent line from the point on the horizon where the island ought to be, if it's not a mirage, to the ridge, from where we saw it, not accounting for refraction. Since it's a tangent, there's a right angle."


"Very nice."

She continued talking, half to herself, as she wrote formulae in the sand. "Pythagorean relationship, hypotenuse, side opposite, side adjacent:"
r2 + d2 = (r + h)2
"I think you're more awake than I am."

"Square the right side:"
r2 + d2 = r2 + 2rh + h2
"Okay."

"Drop the square of the radius from both sides:"
d2 = 2rh + h2
And Bobbie suddenly doubted herself. "Is that right? It looks like it's going to give a number way too large, even after taking the square root."

"The height is much smaller than the radius. Let's plug in some numbers."

"Earth radius is about 4000 miles. But we need it in feet. At five thousand two hundred eighty feet per mile, that's ...," she said as she scratched the numbers in the sand.
  5,280
x 4,000
-------
"Four times eight is thirty-two, ..."
  5,280
x 4,000
-------
 32
"four times two is eight, make it eighty and add the thirty-two, one hundred twelve,"
   5,280
 x 4,000
 -------
  32
  80
 
--------
 112
"four times five is twenty, make it twenty hundred and add the one hundred twelve."
   5,280
 x 4,000
 -------
  32
  80
2000
--------
2112
"Twenty-one twelve. Scale it to a thousand times ten, ten thousand to give ..."
    5,280
  x 4,000
  -------
   32
   80
 2000
---------
 21120000
twenty-one million, one hundred twenty thousand feet for the radius of the earth. How high did you say you thought the ridge is?"

"Six hundred feet. Roughly."

"Twenty-one twelve times six, twelve thousand six seventy-two, slip two more zeroes on the end of that for the hundred:
12,672,000,000
Twelve billion six hundred seventy-two million. Double that:
25,344,000,000
Twenty-five billion three hundred forty-four million."

"Awe inspiring. I'd already lost a column, doing it in my head." 

"Six hundred squared is thirty six times ten thousand, or three hundred sixty thousand. Add that, and the two-r-h term entirely dominates:"
 25,344,000,000
+       360,000
--------------
 25,344,360,000
"Yup."

"I guess this will work. And I dare you to take the square root of that at this time of the morning."

"I think I have my slide rule in my trunk. But let's see if I need it. That's about two hundred fifty-three point four, times a hundred million, which is ten to the eighth, so we can approximate the square root of two hundred fifty-three and multiply that by ten to the fourth, or ten thousand."
(253.4 x 108)1/2 = 253.41/2 x 104



Have you ever seen a slide rule? It's a ruler-within-a-ruler, marked in logarithmic scale, arranged so the inner rule can slide and the marks can be lined up to show approximate products and other arithmetic stuff.

Many include square root scales, to make it easier to multiply by square roots of numbers, and other such time-saving scales.

If you've never seen one, look them up. They're interesting.



"Not bad." But Bobbie didn't wait for Karel. "Fifteen squared is two hundred twenty-five. Add, uhm, fifteen twice and one, thirty-one; sixteen squared is two fifty-six. Oh. Wait. I should've remembered that from the powers of two. Anyway, between fifteen and sixteen times ten thousand, so it's between a hundred fifty and a hundred sixty thousand feet."

"I thought I was supposed to be doing that."

"Divide by five thousand two hundred eighty. Roughly thirty miles out there."

"I am in awe. My wife-to-be will keep me on my toes. Let's go get some breakfast."

Bobbie leaned against Karel's shoulder and said, "I don't get a kiss for that?"

"Do I dare kiss this awesome calculating machine?"

Bobbie answered with a quick poke in his ribs and jumped up. "I'm first in the tent."

"And I'm waiting."

While they took turns changing into their swimsuits, they continued talking about the island.

"Thirty miles. That's not very close," Bobbie said from inside the tent.

"Ten hours of steady, hard rowing, without a lot to navigate by. And, if what we saw was, say, the tip of a mountain, it would be farther than that."

They traded places and Karel changed.

"Do you think it was a mirage?"

"I think that's a possibility."

"We should hike up there and look again. And pray about it."

"I think you're right."

As they walked to the beach, they talked about what they would need to do if they went looking for the island.

"It's going to be really hot out on the water during the day."

"Maybe it would be better to start in the evening?"

"If we do that, we'll want a bright moon so we can see our island behind us as long as possible."

"Ten hours alone with the moon on the waters and nothing around us. Very romantic."

"Lotta temptation, if we go looking before we get married."

Karel stopped, and Bobbie looked back. For a few heartbeats, they just looked at each other, talking to each other with their eyes. What they said was their business, not ours. It probably would have been hard to put into words, anyway.

Karel sighed and continued. "We'll need a lot of water and a couple of baked breadfruit, I think. The menu will be anything but romantic, and we're going to be hungry by the time we get back."

While they were washing clothes, Bobbie mentioned something that had been worrying her lately.

"I missed my monthly last month."

"Huh?"

"You know, the monthly stuff that women go through. We talked about tricks the male body plays on a guy, can we talk about tricks the female body plays on a girl?"

"You're talking about your monthly menstruation cycle."

"Yeah."

"You're the nurse."

"You're my boyfriend."

"Right now I'm your brother."

"Makes it worse."

"Okay, I'm your mom."

Bobbie looked at Karel. "I'm having a hard time believing that."

"Then I'm your husband and we're in a time slip."

"Not logical, but I'll go with it. I missed my period last month."

"That's what I thought you said, now that you mention it."

Bobbie splashed Karel with the blouse she was washing. He ducked, but not fast enough to avoid a face full of salt water.

"Okay. This is important. I'm listening."

"Thank you. My period was supposed to start on the day after we flew out from the last island. Our second day here. It didn't."

"Stress from being kidnapped, I assume? My sisters sometimes talked about stress causing a skip or something."

"Well, I don't remember anything else happening to me that could have caused me to miss."

"I'm not sure what I should say to that."

Again, they talked to each other with their eyes.

"Heh. I guess that could be a kind of a trap. Anyway, stuff is kind of building up inside of me. Skipped cycles don't always mean the next one is going to be hard, but this one feels like it's going to be hard."

"Then you're going to have to let me be your mom, at least."

"If we move the date up a couple of days, you could just be my husband."

"I'll help you anyway I can, Bobbie. I think you know that. Stomach rub, back rub, leg rub, shoulder rub, washing your stuff, moving wedding dates up, whatever."

"Thanks."

After they finished washing their clothes and bathing, they caught some fish and seaweed for breakfast and headed back for camp.
"Once you start spotting, we'll have to keep an extra sharp eye out for sharks, I think. but do you cramp easily in the water when you're in that part of the cycle?"

"Not usually. But we may need to think ahead, and stay out of the water as much as possible."

"I don't think we're going to the island in the next few days."

"We should go up and take a look anyway."

"Yeah."

After breakfast, they read a couple of randomly chosen scriptures and prayed again for guidance about their marriage plans. Then they grabbed their backpacks.

"Journals?"

"Yeah."

For some reason, they both ended up with both their current journals and the notebooks where they had written their marriage license up in their packs.

And they headed up to the ridge.

Arriving at the top, they looked around the island. Unfortunately, clouds obscured the horizon to the east.

"Well, that's disappointing," Karel complained.

"Let's explore. Maybe we can find the highest part of the ridge."

Hiking around the east side of the ridge, they found some new plants and took samples of them. Coming around the west side of the ridge, they found a clearing nestled into the ridge where it would be sheltered from the wind. Surrounded by trees and the ridge wall, they could not see the ocean.

In the center of the clearing, a large, sort of rectangular rock jutted out of the ground, like a tall backless bench. Seeing it, they both sat down on it to rest.

The link to the next chapter will be here when it's ready is here: http://free-is-not-free.blogspot.com/2016/08/economics-101-novel-ch40-choices-and.html.

(The chapter index is here: http://joel-rees-economics.blogspot.jp/2016/04/economics-101-novel-index.html)

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