I wrote something a few weeks ago about measuring the height of Mount Everest. Now you simply use GPS? The article says that its more complicated than that.

Initially, it was the Trigonometric Survey of India in which they marched across the sub-continent, measuring angles angles and finding distance until being able to measure the height of Mount Everest. George Everest, a surveyor, was an important part of the efforts to measure it. 

Some interesting facts that it came out:

• There was a recent earthquake that lowered its height, but it was only by one inch.
• A legend states that the measurement came out to be exactly 29,000 feet. They figured people would suspect a figure like that, so they tacked on a couple more feet so they published the height as 29,000 feet.
• For GPS you need a receiver. There is one near the top. The official height is from a survey done in the fifties.

There are complications I wasn't aware of. The elevation is the height above sea level. But what exactly is sea level. To begin with, the earth isn't completely round. Sea level is not level due to the tides. It is basically the average of the high and low tides, but that gets a little tricky to measure. That can also differ because other mountains in the area can alter the gravity, which then alters the tide height.

As the article stated, "All of our elevations have an error."

 

Wilt's Free Throws

 Wilt Chamberlain is the second leading scorer of all time. Right behind Michael Jordan. But, he couldn't shoot free throws. He once missed 22 straight free throws. That seems worth investigating. First what are the chances? He made 51.1% of his free-throws during his career. In other words he has a 48.9% chance of missing. So to miss two straight would be .489 x .489 = .239121 or a 23.9% chance. Twenty-two in a row? That would be .489 to the 22nd power. That comes out to 0.000000146. 

Just for fun, I took the reciprocal. That is almost seven million. He shot over eleven thousand free throws in his career. I thought missing 22 in a row might be somewhat likely. But no, it isn't. There is a lot of math here a person could play with. Wilt wasn't the worst ever. Andre Drummond makes 38.6% of his free throws. You could figure the probability for him missing 22 in a row.

Could a typical player do this? Lets suppose we want to find out, what percent would you normally have to shoot to have a 50% probability of of missing 22 in a row.  The equation would be x^22 = .50. Using the log and inverse log on your calculator, you come up with 96.9%. That is the chance you miss a shot. So, that means your free throw success rate would be 3.1%. Not good.

Election Results

 There was an important Senate election in Georgia (on the website I was looking at it was "Georiga"). I saw this :

Candidate A   51.5%  -  1,957,641

Candidate B   48.5%  -  1,844,815

That was with 87% of the vote counted.

If the vote is split evenly the rest of the way, Candidate A wins, obviously. What percent would be needed? I figured it was my duty as a citizen to find out.

There are a total of 3,802,456 votes so far with 87 % in. That means there must be a total of:

0.87x = 3,802,456

x = 4,370,639

Candidate A needs at least half of that, which is 2,185,319.

With a little subtraction there are 586,183 votes left and he need 227,678 of them. Candidate A needs 40.1% of the remaining votes. That sounds about right. Certain sections get counted at different times. So its not over. 

Update - Its not decided yet, but Candidate A now has 50.02% of the vote. About as close as it can get.

Increases

 I heard a news report that there were 111 deaths among the homeless due to Covid. This was a 23% increase. Sad news of course, but it made me think of what that meant if you wanted to know what the amount was previously. A little algebra would do the trick.

Assume there were "x" deaths previously, so the equation would be:

                        x deaths plus a 23% increase is 111

                        x + 0.23x = 111        

                         1.23x = 111

                         x = 111/1.23

                         x = 90

A practical, every day application

Measuring Mountains

Here is an interesting item I saw online

"Accurately measuring miniscule changes in a mountain that is more than 5 miles us is no easy feat, but surprisingly, measurements rely on geometric formulas and surveying techniques that haven't changed all that much since the 1800's, said Peter Molnar, a geologist at the university of Colorado, Boulder.

At heart, measuring a mountain relies on basic ninth-grade math. To calculate the elevation of a mountain, scientists would measure the distance. between two points on the ground and then measure the angles between the top of the mountain and each point. 

"if you have two angles, you know the third, because the sum of the angles is 180 [degrees], Molnar told Live Science. 

To carry out these measurements, surveyors must identify a horizontal surface using a level (which, like the kind at a hardware store, relies on a trapped air bubble that, under the influence of gravity, slides closer to or farther away from a central region as it tilts". From there, surveyors eye the summit and measure the angle with the assistance of a glorified highly accurate protractor - a telescopic device known as a theodolite. With two angles and one side of  a triangle, trigonometry reveals the lengths of the other sides, and thereby, the height of the triangle (the mountain)."

If I'm getting the picture they are trying to convey, this method (I assume the Law of Sines) would not give the height. It would give distances from each of the individuals to the top of the mountain. .From there, you could use one of those distances and set up a vertical right triangle and measure the angle of elevation to the summit. Then you could simply use: Sin(measure of angle of elevation) = opposite side / hypotenuse. Solve the equation for  the opposite side and that is the height of the mountain.

There were other interesting things in the article. That can be for another time. 

Virus Probabilities

 I saw something on the news about the coronavirus. Wearing a mask is 70% effective in blocking the droplets that can cause you to be sick. If you and the other person is wearing a mask, you only have a 9% of getting infected. (Those numbers are what I heard, but I'm not certain of the context because I wasn't totally paying attention, but we'll run with it.) Those numbers seem right, mathematically. You have person A and person B. You want either one's mask to block the germs.

Probablility of A or B is happening is 

Pr(A) + Pr(B) - Pr(A) x Pr(B).=  0.7  +  0.7  - 0.7 x 0.7 = 1.4 - 0.49 = 0.91. 

Thus,a 9% chance germs get through.

Looking at it another way, start directly with the probability germs get through.

C and D are now the events that germs aren't blocked.

Pr(C) x Pr(D) = 0.3 x 0.3 = 0.09

Again a 9% chance.

Pr(B) - Pr(A) x Pr(B).=  0.3 +  0.  - 0.7 x 0.7 = 1.4 - 0.49 = 0.91. 

Leaves

 I looked out upon my lawn covered with leaves and wondered how many there were. The brute force method is to go count them. Lets say I don't have the time or inclination to do that. I could take some smaller areas of the lawn, count the leaves there, and compare them to the whole.

This is the same kind of thing that is done in surveys. DVD sales, TV ratings, unemployment numbers, all survey to get their results. We recently had a presidential election. It took a while, a long while, because they said they were getting to a 99.5%. That doesn't give you certainty of a number. It just means you are 99.5% certain of being within a range of numbers. Almost all nationwide statistics are surveys. When we don't do that - the census every 10 years - it takes months and months to get to everyone.

I'm sure you know how this kind of thing works, but I took the time to do it, so I feel obligated to share.

I (roughly) marked one yard by one yard squares. There were three of them and they averaged having 125 leaves in each. Then I found the area of the entire yard. Nicely, it is pretty much a trapezoid. The bases about 7 and 14 yards with a height of 7 yards. That gives an area of 73.5 square yards.

Next, a proportion would be 125/1 = x/73.5. So, x = 9187.5 leaves. 

So, a nice problem for maybe an Algebra I or general math class. It can lead into the idea of the surveys that go on every day.