Can you pass the Phil please?

Q: A plane crashes at the U.S.-Canada border. Where do you bury the survivors?

A: You don’t bury survivors.

But if you were the only survivor, how long could you survive by eating the other passengers?

We’re going to assume it’s winter and the bodies stay frozen so they don’t decay. A Boeing 747 can carry 416 passengers, meaning there are 415 potential menu choices. Assuming our cannibal doesn’t eat the bone, he’ll be getting all of his calories from the rest of the tissue. A normal-sized person is about 68 kg (~150 lbs). Skeletons weigh about 14% of the total body weight, leaving about 59 kg of edible other stuff. Since protein is 4.0 Calories per gram and fat is 9.0 Calories per gram, we’ll assume that the average caloric density is 6.0 Calories per gram. From this we can compute the total Calories in each person,

Cal. per person = (edible mass of person) · (Cal. per g)

= (59 kg) · (6.0 Cal per g)

= 3.5×10⁵ Cal. per person.

Multiplying by 415 people, we can compute the total number of Calories,

total Cal. = (Cal. per person) (# of people)

= (3.5×10⁵ Cal. per person) · (415 people)

= 1.5×10⁸ Cal.

Assuming a 2000 Calorie per day diet, we can compute just how long it would take to eat and successfully digest all the other passengers,

time = (total # of Cal.) / (# Cal. per day)

= (1.5×10⁸ Cal.) / (2000 Cal. per day)

= 200 years.

It would take 200 years to eat all the passengers. So, if you’re the only survivor in a plane crash, look on the bright side. At least you wouldn’t starve to death.

Chris Moore and Vampire Hummingbirds

This question comes courtesy of the very funny and talented Christopher Moore, author of The Stupidest Angel, Fluke, Lamb, and many other hilarious books. Chris wants to know…

“If your body could be reduced to its pure sugar content (sure, you can use equivalent carb calories), how long would it take 1000 hummingbirds to drink you as nectar?”

There are many interesting hummingbird facts on the Web. According to one source:

“Hummingbirds feed on a variety of flower nectars with caloric values that may vary from 10 to 82 calories per meal (1/100 fluid ounces). We found that when using a relative rich sugar solution, a three-gram male Ruby-throated Hummingbird ate five meals an hour.”

Taking the mean, we’ll say that hummingbirds consume 46 Calories per meal or roughly 230 Calories per hour. There are 4.0 Calories per gram in a carbohydrate. From this we can compute the mass of carbohydrates consumed by a hummingbird in an hour,

mass eaten per hr = (Cal. eaten per hr) / (Cal. per g of carbs)

= (230 Cal. consumed per hour) / (4.0 Cal. per gram)

= 58 grams of carbs per hour per hummingbird

This means that 1000 hummingbirds would consume 58,000 grams of carbohydrates per hour.

Figuring out how many carbohydrates are in the human body is a trickier question. Many diet websites suggest getting anywhere from 40-65% of your daily calories from carbohydrates, but the human body doesn’t store anything close to this amount. According to at least one source, the human body is about 61.8% water, 16.6% protein and 14.9% fat, leaving at most 6.7% for carbohydrates. In general, humans store energy in glycogen, also called animal starch, instead of table sugar. I’ll assume this is what the hummingbirds are after. According to Wikipedia, glycogen makes up about 8% (~100-120 g) of our liver, but only about 1-2% or our muscles mass. Muscles make up about 42% of male human body, which can weigh about 82 kg (~180 lbs). From this we can estimate the total mass of glycogen in the human body,

total mass = (mass in liver) + (mass in muscles)

= (110 g) + (0.01)

-->· (0.42) · (82 kg)

= 450 g

Now we can compute the total time it takes for a hummingbird to devour all the carbohydrates in a human,

time = (g of carbs per human) / (g of carbs eaten per hour)

= (450 g per human) / (58,000 g per hour)

= 28 s

It would take only about 28 seconds for hummingbirds to suck all the nectar out of a human body.

This, of course, assumes we have normal hummingbirds rather than vampire chupacabra-eque hummingbirds, which would devour our entire body like a cartoon piranha. How long would it take for 1000 vampire chupacabra-eque hummingbirds to devour a human body? Assuming they eat just as fast as normal humming birds, we can use the same math above,

time = (g of total human) / (g eaten per hour)

= (81 kg) / (58,000 g per hour)

= 1.4 hours

It would take about one hour and twenty-three minutes to be eaten by vampire hummingbirds. Thanks so much to Chris Moore for taking the time to submit a Fermi problem. Check out his very funny books at www.chrismoore.com or follow him on Twitter @TheAuthorGuy.

Estimation Contest Round II

Check out the blog next week for a chance to win a signed copy of How Many Licks?

A Whale of a Time

Fox once aired a show called Man vs. Beast, featuring several of nature's greatest rivalries: world class sprinter Shawn Crawford vs. a zebra, hot dog eating champion Takeru Kobayashi vs. a Kodiak bear, and—my personal favorite—a team of little people competing with an elephant to pull a DC-10 airliner. This type of quality programming inspires mathematical minds. What other great match-ups could we have? For example, how many musclemen would it take to hold back a whale?

Balaenoptera musculus, the blue whale, averages 150 tons (1.5×10⁵ kg) and can reach a top speed of about 31 mph, which is roughly 14 m/s.  Assuming it takes 10 seconds to reach top speed, we can calculate the force the whale uses to accelerate itself using the formula,

force = (mass) · (change in velocity) / (change in time)

= (1.5×10⁵ kg) · (14 m/s) / (10 s)

= 2.1×10⁵ N.

Here, “N” is short for “Newton”, a unit of force. The musclemen would need to exert an equal and opposite force on the whale to hold him back. In the gym, it’s common to see some of the bigger men doing lateral rows with 200 lbs, which corresponds to 980 N.

From this we can calculate how many musclemen we’d need to hold back a whale,

number of musclemen = (force per whale) / (force per muscle man)

= (2.1×10⁵ N per whale) / (980 N per muscle man)

= 214 musclemen per whale.

It would take over 200 fairly strong people to hold back a whale. Check out Diary of Numbers tomorrow, as we’ll have a question from a special guest!

Asteroid Attack!!!

Every few decades, an asteroid comes dangerously close to Earth’s orbit. While an impact with the Earth would surely cause widespread destruction of civilization and an end to life as we know it, this outcome may be preferable to sitting through Armageddon, Deep Impact, Meteor, or whatever is the next in a long line of horrendously clichéd movies about giant space rocks hurtling towards the Earth threatening to, once again, destroy life as we know it. But if an asteroid were about to hit the Earth, could we save our planet by all jumping simultaneously? If everyone on the planet gathered on the streets of New York and jumped, how fast would the Earth be pushed out of the way?

There are 6.7×10⁹ people in the world and people weigh roughly 65 kg (~140 lbs) on average. This means the total mass of all humanity is given by,

total mass = (mass per person) · (# of people)

(65 kg per person) · (6.7×10⁹ people)

= 4.4×10¹¹ kg

If they all jump 0.3 m (~1.0 ft) off the ground, then we can compute the total potential energy from the jump using the formula below,

potential energy = (total mass) · (gravitational acceleration) · (height)

= (4.4×10¹¹ kg) · (9.8 m/s²) · (0.3 m)

= 1.3×10¹² kg.

Using the conservation of energy and the formula for kinetic energy, we can compute the speed at which they would leave the ground,

velocity = [2 · (kinetic energy) / (total mass)]^1/2

= [2 · (1.3×10¹² kg) / (4.4×10¹¹ kg)]^1/2

= 2.4 m/s

People leave the ground with a speed of about 2.4 m/s. The law of conservation of momentum states that when two things push off each other, the momentum of one thing is equal and opposite to that of the other thing. The momentum of an object is equal to its mass times its velocity, so we can compute the momentum of the people,

momentum = (mass) · (velocity)

= (1.3×10¹² kg) · (2.4 m/s)

= 3.1×10¹² kg·m/s.

The Earth’s mass is about 6.0×10²⁴ kg. From this, we can calculate the speed at which the Earth will recoil,

Earth’s velocity = (momentum) / (mass)

= (3.1×10¹² kg·m/s) / (6.0×10²⁴ kg)

= 5.2×10¹³ m/s.

The Earth would only move at about 5.2×10^-13 m/s. At that speed, the Earth would take an hour to move a distance of only a few atoms!

The Great Heater of Buffalo

Much of the northeast U.S. is currently digging its way out from all the snow. Snow is a big pain in many northern states. Upstate New York can get over 2.8 m (~100 in) of snow a year. Why don’t we just run a giant heater to melt it all? How much would it cost to melt all the snow in Buffalo?

The total area of Buffalo, NY is about 2.6×10⁹ m² (~1000 miles²). From the numbers above, we can compute the total volume of snow,

volume = height · area

= (2.8 m) · (2.6×10⁹ m²)

= 7.3×10⁹ m³.

Snow has a density of about 80 kg/m³. From this, we can compute the total mass of snow,

mass = density · volume

= (7.3×10⁹ m³) · (80 kg/m³)

= 5.9×10¹¹ kg.

The amount of energy it takes to melt a solid is called the heat of fusion. The heat of fusion of ice is 334 kJ/kg, meaning it takes 334 kJ of energy to melt 1 kg of ice. From the previous problem, the cost of energy is about 12.56 cents per kW·hrs. From this data, we can compute the total cost to melt one year worth of snow in Buffalo,

cost = (total mass of snow) · (heat of fusion of ice) · (cost of energy)

= (5.9×10¹¹ kg) · (334 kJ/kg) · (12.56 cents per kW·hrs)

= $6.9×10⁹.

At almost $7 billion a year, it would cost about 40% of NASA’s entire budget.

A Dry Calculation

Just finishing up the laundry and wondering, how much money would we save in our lifetime if we used clothes lines instead of electric dryers?¹

There are two ways to calculate this depending on whether or not you use your own electric dryer or one from a laundromat. For the laundromat, it generally costs around $1.25 to do a load of laundry. If you do a load of whites and one load of colors each week, then you spend $2.50 per week or $130 dollars per year. If you live for 80 years, you’ll waste $10,000 in your lifespan.

If you own your own dryer, it’s a little more complicated because you’re paying for electricity. A single dryer load takes anywhere from 30 minutes to one hour, so we’ll assume 45 minutes. If you do two loads each week, you’re running it for 1.5 hours. At least one reference lists a dryer’s power consumption as being about 4.0 kW. From the numbers above, we can calculate the total amount of energy used each week,

energy = (power) · (time)

= (4.0 kW) · (1.5 hours)

= 6 kW·hrs per week.

In Michigan, the cost of electricity is about 12.56 cents per kW·hr. From these numbers, we can compute the cost per week,

total cost per year = (number of kW·hrs per week) · (cost per kW·hr)

= (6 kW·hrs per week) · (12.56 cents per kW·hrs)

= 75.36 cents per week

= $39.18 per year.

Using the same lifespan as above, you would save about $3100 dollars in electric bills over a lifetime. In addition, you would save money by not purchasing a dryer in the first place. A quick Google product search shows that electric dryers cost around $400. Dryers have to be replaced about once every 8 years. This means you’d need to buy about 10 dryers in a lifetime, costing an extra $4000 dollars bringing the total savings to about $7100.

In short, dry your clothes on a line when you can. It’s good for the environment and it can save you thousands of dollars.

[1] While researching this problem I can across a very good site here.