## Posts tagged ‘distance’

### Guess the Graph on Square Root Day

Today is Opening Day in Major League Baseball, and 13 games will be played today.

It’s also Square Root Day, because the date 4/4/16 transforms to 4 × 4 = 16.

With those two things in mind, here’s a trivia question that seems appropriate. Identify the data set used to create the graph below. I’ll give you some hints:

• The data set contains 4,906 elements.
• It’s based on a real-world phenomenon from 2015.
• The special points marked by A, B, and C won’t help you identify the data set, but they will be discussed below.

Got a guess?

No clue? Okay, one more hint:

• The horizontal axis represents “Distance (Feet)” and the vertical axis represents “Frequency.”

Still not sure? Final hints:

• Point A on the graph represents Ruben Tejada’s 231-foot inside-the-park home run on September 2, 2015.
• Point B on the graph represents the shortest distance to the wall in any Major League Baseball park — a mere 302 feet to the right field fence at Boston’s Fenway Park.
• Point C represents the longest distance to a Major League wall — a preposterous 436 feet to the deepest part of center field at Minute Maid Park in Houston.

Okay, you’ve probably guessed by now that the data underlying the graph is the distance of all home runs hit in Major League Baseball during the 2015 season. That’s right, there were 4,906 home runs last year, of which 11 were the inside-the-park variety. The distances ranged from 231 to 484 feet, with the average stretching the tape to 398 feet, and the most common distance being 412 feet (86 HRs traveled that far). The data set includes 105 outliers (based on the 1.5 × IQR convention), which explains why a box plot of the data looks so freaky:

The variety of shapes and sizes of MLB parks helps to explain the data. Like all math folks, I love a good graphic, and this one from Louis J. Spirito at the Thirty81Project.com is both awesome and enlightening:

click the image to see the full infographic

Here are some more baseball-related trivia you can use to impress your friends at a cocktail party or math department mixer.

1. Who holds the record for most inside-the-park home runs in MLB history?
Jesse Burkett, 55 (which is 20 more than he hit outside-the-park)
2. Which stadium has the tallest wall?
The left field fence at Fenway Park (a.k.a., the “Green Monster”) is 37 feet tall.
3. Which stadium has the shortest wall?
This honor also belongs to Fenway Park, whose right field wall is only 3 feet tall.
4. Although only 1 in 446 home runs was an inside-the-park home run in 2015, throughout all of MLB history, inside-the-park home runs have represented 1 in ____ home runs.
158
5. Name all the ways to get on first base without getting a hit.
This is a topic of much debate, and conversations about it have taken me and my friends at the local pub well into the wee hours of the morning. I have variously heard that there are 8, 9, 11, and 23 different ways to get on base without getting a hit. I think there are 8; below is my list.
(1) Walk
(2) Hit by Pitch
(3) Error
(4) Fielder’s Choice
(5) Interference
(6) Obstruction
(7) Uncaught Third Strike
(8) Pinch Runner
6. What is the fewest games a team can win and still make the playoffs?
39. The five teams in a division play 19 games against each of the other four teams in their division. Assume that each of those teams lose all of the 86 games against teams not in their division. Then they could finish with 39, 38, 38, 38, and 37 wins, respectively, and the team with 39 wins would make the playoffs by winning the division.
7. Bases loaded in the bottom of the ninth of a scoreless game, and the batter hits a triple. What’s the final score of the game?
1-0. By rule, the game ends when the first player touches home plate.
8. In a 9-inning game, the visiting team scores 1 run per inning, and the home team scores 2 runs per inning. What is the final score?
16-9. The home team would not bat in the bottom of the ninth, since they were leading.

### Lighthouses and Math in Oregon

My family and I recently spent a week along the Oregon coast, where we hiked, biked, and — of course — did some math.

While on a hike through Ecola State Park, we had this view of the Tillamook Rock Lighthouse, a structure 62 feet tall that stands atop an offshore island:

Never one to resist the opportunity to pose a math question, I asked my sons, “How far do you think it is from here to the lighthouse?” My sons — never ones to resist reading the informational placard at a trailhead — already knew the answer and quickly responded, “About 1.2 miles.”

Ah, hell no. I’m itching for a mathy conversation, damn it, and you’re not getting off that easy! So I then asked, “How far do you think it is from here to the horizon?” Luckily, the answer to that question hadn’t been included in the display at the bottom of the hill.

Eli used proportional reasoning. He extended his arm and measured the distance from the shore to the lighthouse between his thumb and index finger; then, he used that same unit to measure the distance from the lighthouse to the horizon. “I’d guess about 2 miles,” he said. “It’s farther from here to the lighthouse than from the lighthouse to the horizon.”

Alex used intuition. “I think it’s a little farther,” he said. “Maybe three miles?”

“Actually,” I said, “it’s much farther than that. I won’t tell you how much farther, but I will help you figure it out.” (Typing that last sentence makes me realize that having a constructivist father is not an easy burden to bear. My answer to most questions is usually another question.)

After much discussion, we finally concluded that we were about 90 feet above sea level. Estimating vertical distance, it turns out, is not an easy task, especially for eight-year-olds. But once we agreed, the rest of the math was more or less straightforward. The image below provides a reasonable model for the situation:

In that model,

• h = height above sea level of the observer
• R = radius of the Earth
• x = the straight-line distance to the horizon from the observer

The sight line of the observer is a tangent line to the circle, and the point of tangency is the horizon that the observer sees. Consequently, a right angle is formed by the tangent line and radius. We now have a right triangle with which to work, so we can lean on the Pythagorean theorem. But first we’ll need some numbers.

At the Tillamook Rock Lighthouse, we had the following numbers:

• h = 90 feet (estimated)
• R = 4,000 miles (give or take)

And x, of course, is the unknown distance to the horizon that we are trying to find.

$\begin{array}{rcl} x^2 + R^2 & = &(R + h)^2 \\ x^2 + 20\,000\,000^2 & = &(20\,000\,000 + 100)^2 \\ x^2 & = &40\,000\,010\,000 \\ x & \approx &62\,000 \, \textrm{feet} \end{array}$

A rough estimate of the Earth’s radius in feet is 4,000 × 5,000 = 20,000,000, which is why that number appears in the equations above. Two things should be noted here. First, using 5,000 for the number of feet in a mile instead of the more accurate but less friendly 5,280 allows for easier calculations throughout. Second, we’re only looking for an estimate of the distance to the horizon, so we should feel free to take some liberties with the numbers if it’ll help with computation.

Using 20,000,000 instead of 4,000 allows feet to be used as the units throughout, so the answer obtained at the end (x = 62,000) is also in feet. To find the distance to the horizon in miles, we’ll need to divide by 5,000, which gives:

62,000 ÷ 5,000 = 12.4 miles

Now that’s all well and good, and the boys were amazed that the horizon is so far away. But what if you want to generalize to find the distance to the horizon from any height above sea level?

The answer depends on the math, boating, or physics book you choose to reference.

One possibility is to use this graph, which we discovered a few days later while visiting the Yaquina Head Lighthouse:

Graph at the Yaquina Head Lighthouse, Newport, OR

It shows the distance to the horizon on the horizontal axis and the feet above sea level on the vertical axis, which I suppose makes sense from a visual perspective; but it’s more conventional to put the independent variable (feet) on the horizontal axis and the dependent variable (miles) on the vertical axis. Nonetheless, the information provided by the graph is still useful, and it nicely shows that the relationship between height and distance to the horizon is clearly not linear.

On the other hand, if you search for “distance to horizon formula” online, you’ll find any of the following formulas:

$\begin{array}{rcl} d & = &\sqrt{1.5h} \\ d & = &\sqrt{\frac{7h}{4}} \\ d & = & 1.17\sqrt{h} \\ d & = &R\times\arccos\bigl({\frac{R}{R+h}\bigr)} \end{array}$

The differences are due to whether miles or nautical miles are used, as well as whether you’d like to account for the refraction of light, which greatly influences the distance an observer would be able to see. Personally, I prefer the first formula for its simplicity, but you’re free to use whichever one you like.

Sadly, I don’t know a single joke involving the horizon. (Do you? Post in the comments.) So I’ll leave you instead with a quote about the horizon:

Make the horizon your goal. It will always be ahead of you.
— William Makepeace Thackeray

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

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