A Ton Of Snow

The other day my students were taking their pre-calc exam and due to my writing it too long, many needed to stay after class was over to finish. So many in fact, that I emailed "Wow, I have a ton of students needing to stay after!" to my principal.

After hitting send, I was instantly annoyed that I had used "ton" as an exaggeration -- something I have been known to chide my students about. But I was comforted when I did a quick estimation and realized that I was actually fairly accurate:

As it turned out, I had 14 students in my room, and many of them were the bigger athletes (certainly bigger than 120 lb) so I'm sticking to my original statement as literally true. And is 120 lb an appropriate average weight of a student?  Who knows.

Well, today I was out shoveling snow for the first time this season, and I caught myself again saying "This is a ton of snow!"  Well, was it? Was it really?

To calculate the weight of snow, I need to find the volume of snow on my driveway, and multiply it by the density of snow.  The volume of snow is easy enough to approximate -- I'll assume I have an even rectangular driveway with the same height of snow everywhere, and so volume of this rectangular prism is just length * width * height:

 The density of snow is a little more difficult, as there is heavy snow, light fluffy snow, solid ice, etc.  Wikipedia says snow has a density of anywhere from 8% - 50% of water, depending on many things, but mostly on how compacted it is as it melts, freezes, melts, refreezes and more snow falls on top of it. This was relatively new snow, and so I'll assume it has a density of about 25% of water.

The density of water is 1 kg/L.  This "coincidence" of having such a clean number is actually not a coincidence at all, but was by design -- as 1 kg was originally defined to the weight of 1 liter of water as the metric system was being invented. Since then we have defined the kg more precisely than that using more complicated methods. Though it's no longer exactly 1, this value remains accurate enough for our purposes (I think it's 1.003 or something close?)  Converting kg to pounds and L to cubic inches is a tough exercise:

If my snow was 25% the density of water, than:

The weight of the snow on my driveway is then:

Looks like I wasn't exaggerating after all! I moved over a ton of snow today!

How much daylight is left?

In yesterday's post, we learned about the angle, the foundation measurement tool for observing things in the sky. Today you'll use those tools, and your hand, to get a quick estimate of the amount of daylight left.

First, we should figure out how quickly the sun moves across the sky. Because the earth rotates once every 24 hours, the sun appears to move across the sky once every 24 hours. To be more specific:


Since the width of your hand spans approximately 10-15 degrees (mine is relatively "fat" and covers 15), you can use it to approximate how far the sun will move in an hour. So, what I've done on many occasions is counted how many hands up from the horizon the sun is, and approximated how long till sunset. Since you have 4 fingers, they make decent 15 minute approximations.

A few notes -- first, the sun does not travel straight down, but at an angle towards its final resting place. In the Northern Hemisphere (specifically North of the Tropic of Cancer line) where I'm guessing any of you readers are from, the sun will move further north as it sets, so you might need to tilt your hand somewhat to accommodate.

Secondly, it doesn't become instantly dark once the sun sets, but there is plenty of twilight to help you. I typically figure on an additional hour of twilight before it gets too buggy or dark to want to be outside.

If we know the width of the sun, we can calculate how long sunset will take, from the moment the sun first touches the horizon till it dips behind the horizon. The "width" of the sun, and the moon for that matter, is about 1/2 degree. So once the edge of the sun touches the horizon, you'll have: to enjoy the sunset before its gone.

You can also use this fact if your clock on your camera ever goes bad to tell when a picture was taken. In photoshop (who am I kidding... i just used MsPaint, I can't afford photoshop) I projected where the sun was going to travel, and more importantly how many "suns" were left in the sky. Since five suns were left, this picture was taken approximately 10 minutes before "sunset" which you can look up for any particular place and day.

Work-Cost of Activities

Inspired by a quote on Dave Ramsey, and a comment by my wife, I thought I would add a little bit to the post on the cost of gas to get somewhere and the money it costs to accomplish various tasks.



As I was reading advice that dad's gave their kids about money, I came upon the quote:

Andrew: We went to a carnival, and afterwards my dad said, "That was the cost of mowing the lawn. Was that worth mowing the lawn?" Now I always think of purchases in terms of hours of work. Is it worth it?

Later, I noticed my wife (who has a blog of her own that I am psuedo-competing with--but don't tell her!) was kind enough to read and comment on one of my posts and said:

So you'd need to make $10 working in GR to make the trip break even. That's why I didn't like summers working at FCS where I'd be scheduled for 2-3 hours and make $20. Half of my paycheck would be going toward gas!

They got me thinking about figuring out how many hours of work does it cost to earn the right to do certain things.

So, here's a handful of calculations, based on an hourly rate of $10. Do I really make $10 per hour? I suppose that depends on how many hours you divide my salary into, but I'm not going to reveal that all, and $10 is easy to divide, so I'll stick with it.

How much work do I need to do to take my wife out to the movies?  Assuming the new $10 price per ticket, $9 for popcorn, $15 for babysitting, and the gas out there and back ($9.60) we're looking at basically 55 dollars: 

   

To fill up the gas tank?
   


To come into work?
   

To buy that iPad you've always wanted?
   

How much work for that Starbucks Mocha Frappucino:
   



A different way to think about it is to figure out how long it takes to earn a dollar, and than start referring to dollar bills in those terms instead.  For me that is:
    
So now I think: wow! Gas costs almost 24 minutes of work a gallon!  

What's the real price of fresh baked bread?

My wife loves making bread with the little bread maker we own.  The other day she pondered making bread so we wouldn't have to buy loaves anymore, and so naturally, I thought I needed to use this as a math post, and see if it's economically sound.

For our two pound recipe, the ingredients include:
   Water: 1.25 cups + 2 TBL
   Oil: 2 TBL
   Sugar: .25 Cup
   Salt: 2 tsp
   Dry Milk:  2 TBL
   White Flour: 3.5 cups
   Wheat Flour: 0.5 cups
   Active Dry Yeast: 2.25 tsp

Except water, the items can be bought at the store "in bulk" and so I researched the prices online:
   Water: ?? Assumed to be free - perhaps a later post will calculate that?
   Oil $3.19 / 48 oz bottle
   Sugar: $2.54 / 5 lb bag
   Salt: $0.49 / 26 oz
   Dry Milk $6.49 / 25.6 oz box
   White Flour $1.49 / 5Lb bag
   Wheat Flour $3.49 / 5Lb bag
   Active Dry Yeast: 4.99 / 4oz jar
 
Converting each of the following into prices per recipe will prove to be a bit challenging, as many items were sold by weight but used by volume. There is no single direct conversion between these two types of measurements, because the same volume of flour will weigh less than the same volume of water, because it has a lighter density. I was tempted to use fluid ounces as a common unit for all items, but because of their different densities, this would incorrect. In fact, flour is so much lighter than water that it would be nearly nearly 2 times more costly to naively assume that (8 / 5.5 to be more precise).

So I will demonstrate the calculation for flour:
     
Using similar logic for the rest of the items, I got the following prices*:
     
     
     
     
     
     

This brings the grand total for a recipe of bread to: $1.33.  Unfortunately this does not yield the same usefulness in terms of number of servings, nor in shelf life, as a typical $1.50 loaf of bread, nor does it include the cost of water (probably negligible?) nor the price of electricity to run the bread maker (probably not negligible).  But it sure tastes good, and makes the house smell a lot prettier than a bag of bread.
And now you know how to make the same sort of calculations for your meals. Anyone want to calculate the true price of lasagna now?

Before you go out and buy your own bread maker, remember that ours was a gift so I didn't include that price in the mix, but if you buy your own, you'll have to estimate how many times you will bake bread and figure in that cost too.  At $100, if you bake 400 loaves you'll add another $0.25 per loaf to the cost.

*Notice several conversions with ounces are different because I looked up the weights of various ingredients.

How many blades of grass do I own?

While mowing the grass the other day, I began thinking about and estimating the number of blades of grass I own. While most of this grass is crab grass, I'll attempt a loose estimation anyway, using unit conversion techniques.  First, a few assumptions.

• My yard is roughly rectangular, approximately 200 feet long and 80 feet wide.  
• A square inch of grass contains on average 100 blades of grass

The area calculation is simple enough, 200 * 80 = 16000 square feet. Converting to square inches requires a multiplication of 12*12 or 144 square inches per square foot.  Altogether then:


Being rather resentful of this grass that continues to require cutting, I also calculated how many grasses get severed per second, knowing that it takes me approximately 75 minutes to mow:
   

How much does it cost to drive to work?

A question few of us think about it on my mind a lot more recently -- especially as the summer approaches. I often will do school work during the summer -- which in most cases I can do from anywhere with a laptop and internet access. Some of that is planning, reorganizing, and some of that is helping build the schedule for next years classes. As nice as it is to be home during the summer, I often work more efficiently from school, simply because I am focused and don't have to worry about distractions. But what does it cost to drive in?

Budgeting has become a monthly chore for our household and one thing we have tried to do is plan our driving to and from Grand Rapids -- about a 30 mile hike one way. This post describes how we make these calculations.

A few assumptions have to be made, or we'll kill each other squabbling over the details. The assumptions I proposed were:

• Distance to work: 30 miles one way or 60 miles round trip
• Miles Per Gallon: 25 in my Ford Taurus
• Price of Gas: $4.  A little high, but I'd rather over budget than under budget, and I also not considering things like oil changes, wiper fluid refills, etc.

This sort of calculation is easily performed using my favorite mathematical tool -- Unit conversion by multiplying by unit fractions. The process is to start with some given quantity and convert it to a desired unit in one chain of fractions. Every time you convert from one unit to another you must either multiply or divide, and in this technique the units are your guide for which operation to perform. When you write each fraction down, you must be sure to write it in a way that cancels out the last unit you had. Specifically, if the units you currently have are in miles on top, the next fraction I should write will have to have miles in the bottom, because a unit in the denominator cancels out a unit in the numerator.

So, lets get started. I'll begin by considering how much it costs to make one round trip, and so I'll write:

The next factor I need to include is the distance of one trip. This can be a fraction that I can include in one of two different ways:

  

The first option represents needing to divide by 60, and the second option represents multiplication. Which operation do I perform? I need to write the second operation, because the roundtrip that is in the numerator will cancel with the round trip in the denominator.  Continuing, I need to write the next conversion factor, 25 miles per gallon in such a way that miles is in the denominator to cancel out miles, so it looks like:

Finally, including the price of gas so that gallons cancels out suggests writing:

Ultimately, any numbers that are on top need to be multiplied, and any on the bottom need to be divided, so this chain of fractions suggests the cost of a round trip is 60 * 4 / 25 = $9.60.

Now rather than guessing at how much money we're going to spend on gas, we can plan on which days we'll be going into Grand Rapids, and calculate how much we'll have to spend. For the month of June, we anticipated 12 trips into Grand Rapids - four mandatory trips for school, 2 for doctors appointments, 4 for softball, and 2 days of open houses. This alone will cost $115 (12*9.60). Sparing you the remaining details regarding shorter trips around town and to the lakehouse, we ended up assuming we would spend $200 on gas.

A few remaining related calculations, just for fun:

• How much does it cost per mile:     

• How far does a penny get you:      which is about the length of a soccer field.  Perhaps we should call pennies "soccer field gassers"? 

• How much would it cost to drive to the moon?