Speed of Light

Today I learned that the first calculations on tge speed of light came when someone noticed that the eclipses cased by Jupiter's moons were appearing several minutes later then the were predicted to. I must research this further to see if there is a way i can turn this into a classroom problem.

Multi-part geometry problem

One side of a right scalene triangle is 100. The perimeter is 240. A similar triangle has a scale factor of 3:5 (new:old). Find the perimeter of the new triangle. Find both possible sets of solutions for the other two sides of the new triangle.

The Math of "The Martian" Sol 98

These are the first two messages that Watney received today. Translate them from ASCII:

Message 1:

434e48414b52565232544c4b32505448464452505250344c4f4e474d5347

Message 2:

4c4e4348686578696469744f4e525652434d502c4f50454e46494c452d2f7573722f6c69622f686162636f6d6d2e736f2d5343524f4c4c54494c4944584f4e4c465449533a32414145352c4f565257525431343142595453575448444154415745274c4c534e444e58544d53472c5354414e44494e56494557344e45585450494332304d494e41465452544853444f4e45

The Math of "The Martian" Sol 97 (2) Solution

Sol 97 (2) SOLUTION

Mark instructed NASA to time their transmissions so they arrived at the top of every hour, with their first transmission arriving at 11:00am. Which means the last transmission from NASA arrived at 5:00pm. Marks’s first note would have to be placed up at least 32 minutes before 11:00 am, so this entire conversation took at least 6 hours and 32 minutes.

ASCII conversions:

STATUS: 535441545553

HOWALIVE: 484F57414C495645

CROPS?: 43524F50533F

WESAW-SATLITE: 57455341572D5341544C495445

BRINGSJRNROUT: 4252494E47534A524E524F5554

SJRNRNOTRSPND: 534A524E534E4F545253504E44

WORKINGONIT: 574F524B494E474F4E4954

The Math of "The Martian" Sol 97 (2)

Sol 97 (2)

Mark has fixed the Pathfinder lander to communicate with earth! Although it will be tedious, the story states that earth and Mars are currently 16 light minutes apart. He can hand write messages on note cards and place them in view of the camera, getting a response is much more tricky. Mark has decided to tell NASA to use ASCII to spell messages. He placed then numbers 0 through 9, and the letters A through F in a circle around the lander. NASA can select one by pointing to the camera at it. The following is their’ first day’s worth of communication:

Mark: Spell with ASCII. 0-F at 21-degree increments. Will watch camera staring 11:00 my time. When message done, return to this position. Wait 20 minutes after completion to take picture (so I can write and post reply). Repeat process at top of every hour.

NASA: STATUS

Mark: No physical problems. All Hab components functional. Eating ¾ rations. Successfully growing crops in Hab with cultivates soil. Note: Situation not Ares 3 crew’s fault. Bad luck.

NASA: HOWALIVE

Mark: Impaled by antenna fragment. Knocked out by decompression. Landed facedown, blood sealed hole. Woke up after crew left. Bio-monitor computer destroyed by puncture. Crew had reason to think me dead. Not their fault.

NASA: CROPS?

Mark: Long story. Extreme botany. Have 126 m2 farmland growing potatoes. Will extend food supply, but not enough to last until Ares 4 landing. Modified rover for long-distance travel, plan to drive to Ares 4.

NASA: WESAW-SATLITE

Mark: Government watching me with satellites? Need tinfoil hat! Also need faster way to communicate. Speak&Spell taking all damn day. Any ideas?

NASA: BRINGSJRNROUT

Mark: Sojourner rover brought out, placed 1 meter due north of lander. If you can concoct it, I can draw hex numbers on the wheels and you can send me six bytes at a time.

NASA: SJRNRNOTRSPND

Mark: Damn. Any other ideas? Need faster communication.

NASA: WORKINGONIT

Mark: Earth is about to set. Resume 08:00 my time tomorrow morning. Tell family I’m fine. Give crew my best. Tell Commander Lewis disco sucks.

How long did this conversation take?

Convert NASA’s transmissions back into ASCII.

The Math of "The Martian" Sol 83 Solution

The Math of "The Martian" Sol 83

Mark is retrieving the Pathfinder space probe in hopes of fixing it’s radio so that he can communicate with earth. He is at Pathfinder’s landing site, but he needs to get the probe onto the roof of his rover. He has removed several parts of the probe that he won’t need, but it still has a mass of 200 kg. How much would that weigh on Earth? How much would that weigh on Mars?

He has decided that the best way to get it up to the top of the rover is to use rocks to build a ramp. After some small scale experiments he has determined that the best angle for the ramp will be 30 degrees. If the rover is 2.5m tall, how long will the base of the ramp be?

If the ramp is 1m wide, and the rocks in the area have a density of 3 g/cm^3, what is the upper limit of the mass of rocks must he move to create his ramp? How much does this weigh on Mars?

The Math of "The Martian" Sol 71 Solution

classdojo.com

The teacher I sub'ed for today used classdojo.com to keep track of class behavior. IT'S AMAZING! I wish all of the teachers I sub for used it!

The Math of "The Martian" Sol 71

Mark is planning on testing his long distance modifications to the Rover with a long distance trip. 

The Hab is located at 31.2 degrees north, 28.5 degrees west. He plans to go to 19.13 degrees north, 33.22 degrees west, and back. How far is that? How many Sols will the trip take? 

The Rover’s Oxygen tanks hold enough for 7 Sols. If Mark needs 588 Liters of gaseous Oxygen per Sol, how much liquid Oxygen does he need to make up the difference? If the Hab has two 25 Liter liquid Oxygen tanks, will he need to take one or both?

His potatoes aren’t ready to harvest yet, so he will need to take ration packs with him. What is the minimum amount of packs that he will he need to take?

The Math of "The Martian" Sol 68 Solution

The Math of "The Martian" Sol 63 Solution

The Math of "The Martian" Sol 40 Solution

The Math of "The Martian" Sol 37 Solution

The Math of "The Martian" Sol 68

Not using the heater didn’t work out. The Rover doesn’t have enough insulation to keep the body heat that Mark generates from leaking out of the Rover. 

Mark’s new plan is to dig up the Radioisotope Thermoelectric Generator (RTG) that powered the MAV fuel plants until the crew arrived and connected the MAV to power form the solar panels.

The RTG turns heat form naturally radioactive Plutonium-238 into electricity. The RTG has two layers of shielding to protect the crew from the radiation produced, as long as both don’t crack it is safe. None the less, once the MAV was connected to the solar panels the crew took the RTG 4 kilometers away from the Hab and buried it.

It’s also not that efficient, the Plutonium inside generates 1500 Watts of heat, but it is only able to convert that into 100 Watts of electricity. But Mark wants it for the heat anyways.

How long would it take to heat 5000 Liters of air, the approximate volume inside the Rover, from -55C, the overnight low on Equatorial Mars, to 20C, a comfortable temperature?

To control the temperature Mark cut large portion of the Rover’s insulation off the wall. He can remove it when the Rover gets too hot, which will allow the heat to escape faster than the RTG can generate it. He can put it back up when it gets to cold, which will allow the RTG to generate heat faster than it leaks out.

Question 1

The Math of "The Martian" Sol 63

Mark states that it takes between 4 and 20 minutes for a signal from Earth to reach Mars. Verify the math behind this statement.



Mark is starting to plan long term, assuming he can find a way to fix the problem with his food supply, he will eventually need to travel to the Ares 4 landing site, 3200 kilometers away, about the distance from Los Angles to Chicago. He will need to modify one of his two rovers to be able to make the drive. He will need to do a lot of calculations to figure out how to make a rover keep him alive for a drive that long, so he is starting with the problem of power.

Each rover has a 9000 watt-hour battery. He plans to transfer the battery from Rover 1 to Rover 2, giving him 18,000 watt-hours per charge. At maximum efficiency the Rovers use 200 watt-hours per kilometer, but he estimates that given the rough terrain he will need to cover, and the extra weight for supplies, it will probably use about 280 watt-hours per kilometer. How far can he go on a charge? How many times will he need to recharge the batteries on the trip to Ares 4?

He will need to recharge the rover using solar panels that are currently powering the Hab. So he will need to drive at night, and charge the rover during the day. The Hab has 50 solar panels, each one has an area of 2 square meters. Mark plans to take 14 for the trip. Each one converts sunlight with an efficiency of 10.2%. If Mars gets 500 Watts of sunlight per square meter, how long will it take to recharge the battery?

If the Rover travels at 25 kph how long can the rover go on a single charge?

How many Sols will it take to drive to Ares 4?

Of course all this is assuming that he doesn't use the heater...

The Math of "The Martian" Sol 40

Mark has a plan for getting rid of the excess Hydrogen in the Hab:

First he will move the live potato plants to the Rover to keep them safe. Next he will drop the temperature in side the Hab down to 1C, this will cause the bacteria growing in his soil to go into hibernation which will greatly decrease the amount of Oxygen they need to survive. Then he will trick the Hab's atmospheric regulator into bringing the Oxygen content in the air down to 1%. This will still be high enough for the bacteria to survive in their hibernation, but to low for the Hydrogen to ignite. Mark will breathe air from one of the spacesuit's supply tanks, and he will use another to add bursts of Oxygen to the air in the vicinity of an electrical arc made from the wires of one of his flash lights. This will allow him to slowly burn off the excess Hydrogen in the Hab without it all igniting at once.

Or so he thinks...

Mark has forgotten to account for one thing. Humans do not use all the oxygen in the air we breath. The air we exhale is still 15% Oxygen, as opposed to 20% Oxygen when we breath in. If the average person exhales 0.5 Liters per breath, and breathes about 20 times per minute under moderate exertion, how long until the Oxygen content of the Hab rises to 5%, enough for all of the Hydrogen to ignite at once?

The Oxygen Mark exhales might not evenly disperse through out the Hab immediately, so the concentration in Mark's vicinity might rise to 5% faster then the rest of the Hab does. To account for this, calculate how long it would take to increase the oxygen content of a 3 cubic meter volume from 1% to 5%.


And indeed the Hydrogen does explode after Mark exhales enough Oxygen into the Hab for it to ignite. Fortunately, Mark had already burned off a lot of Hydrogen in controlled burns before this occurs, also, there probably wasn't enough Oxygen for all of the remaining Hydrogen to react. None the less, the explosion was powerful enough to launch him across the room singe his clothing and cause minor injuries, but not powerful enough to kill him or damage anything important in the Hab.

After reviewing the logs of the atmospheric monitors, the event showed a momentary spike in pressure, as the explosion occurred, but then dropped back down to normal once all the Hydrogen and Oxygen combined into water. It also showed that the temperature in the Hab jumped from 1C to 15C and then stayed there. How much energy would be required to raise the temperature inside the Hab by 14C? How many moles of Hydrogen reacting would be required to release that amount of energy? How many Liters of water was created in this explosion?

Star Method for Factoring Trinomials

A student showed this to me the other day. It seems to work, but I can't prove, or find a proof for it. So if it is used, the student would need to re-foil the binomials at the end to prove they have the correct answer.

Conic Section Summary Sheet

Parabola Summary Sheet

An Alliterative Mind

Problem: define the series and find the next term: 2,3,5,7,11,13
Student: I give up
Me: what are the numbers that can only be divided by one and themselves?
Student: Special Numbers?
Me: Starts with a ‘p’
Student: Pretentious? Progressive? Pragmatic? Preserving?

Using Order of Operations to memorize properties of exponents and logrithims

Beating the Calculator

Me: What’s 18+7?
Student: *reaches for calculator*
Me: *before student finishes punching it in* It’s 25
Student: How do you beat the calculator all the time?
Me: What do you have to steal from seven to get 18 to a multiple of 10?
Student: 2
Me: Yep, what are the numbers now?
Student: 20 + 5?
Me: Yes!
Student: Wow!
Me: Think about that the next time you see someone bashing common core on Facebook. Every single common core rant I’ve seen on facebook shows the ranter’s complete ignorance of the deeper workings of arithmetic.

The Math of "The Martian" Sol 30 SOLUTION

Completing the Square

Methods of solving quadratics

Yesterday I discussed with one of my students when to use each of the three methods to solve quadratic equations: ax^2+bx+c=0…

FACTORING: might not work, easiest if a=1.

COMPLETING THE SQUARE: always works, easiest if b/a is an even integer.

QUADRATIC EQUATION: always works, always difficult.


Also, if the square root goes away in either completing the square or the quadratic equation, then you could have factored.

The Math of "The Martian" Sol 25 SOLUTION

The Math of "The Martian" Sol 16 SOLUTION

The Math of "The Martian" Sol 14 SOLUTION

The Math of "The Martian" Sol 7 SOLUTION

The Math of "The Martian" Sol 37

Mark has a very big problem…

The process for creating extra Oxygen worked perfectly. He released the CO2 from the MAV fuel plant into the Hab. The HAB’s oxygenator pulled the excess CO2 out, converted it to solid Carbon, and liquid Oxygen, and stored both.

Creating Hydrogen worked according to plan as well. Mark set up a fume hood to contain the Hydrazine decomposition reaction. The idea was the released hydrogen would float to the top of the fume hood, where a small tube would lead it up to a small flame. This flame would ignite the hydrogen, which would then combine with the oxygen to create water.

The Hydrazine decomposed just fine, the problem came from the last step, a lot of the hydrogen bypassed the flame and didn’t burn.

The HAB’s atmosphere is now filled with hydrogen. The atmospheric processor won’t pull it out of the atmosphere because an excess of hydrogen was such a remote possibility that it just wasn’t designed to even detect hydrogen, much less remove it.

Mark reacted enough Hydrazine to make 130 Liters of Water by the time he noticed the problem. The water reclaimer pulled 70 liters out of the air and stored it in its tank. Some water is still in the air as Humidity, and some soaked into the soil. But if we ignore those two effects, and assume the entire missing 60 Liters was never made in the first place, what is the upper limit of un-reacted Hydrogen that is now in the Hab atmosphere?

How much enthalapy would be released if it all ignited at once? What would be the equivalent amount of TNT to generate the same release of enthalapy?

Mark took an air sample from the Hab out to one of the Rovers that has a better atmospheric analyzer then the Hab. It said that the Hydrogen composition of the Hab air is 64% Hydrogen by Volume. If the Hab is at least 276 cubic meters in volume, what is the lower limit on the amount of Hydrogen in the air?

How much enthalapy would be released if it all ignited at once? What would be the equivalent amount of TNT to generate the same release of enthalapy?

The Math of "The Martian" Sol 30

Mark needs to make water. As he says “fortunately I know the recipe: Mix 2 parts Hydrogen with 1 part oxygen and burn”

Mark plans to get the Hydrogen from The Mars Decent Vehicle (MDV). The MDV is a small space craft that was only designed to take them from their main spaceship, the Hermes, in orbit, to the surface of Mars. It uses Hydrazine (N2H4) to power its thrusters. Because Martinez, the mission pilot, did such a good job on his landing, the MDV has 292 Liters of Hydrazine left in the fuel tanks. Almost all of the Hydrazine can be broken down into N2 and H2 using the Iridium catalyst in the MDV’s thruster reaction chamber. Not all of the Hydrazine will convert to N2 and H2, a small portion will become ammonia.

If however we assume that all the Hydrazine is converted, then we can get an upper limit on how much water Mark can make. If Mark had an unlimited supply of Oxygen, how many Liters of water could he make with the Hydrazine he has, again, assuming all the Hydrazine reacts? What is the highest percentage of Hydrazine that can turn to ammonia will still yielding the amount of water Mark needs?

How much energy would be released if all the Hydrazine was converted to Nitrogen and Hydrogen in the absence of Oxygen all at once? I.E. If the hydrazine exploded outside the Hab. Express your answer in terms of Enthalpy. How many pounds of Trinitrotoluene (TNT) would be required for the same release of Enthalpy?

How much energy would be released if all the Hydrazine was converted to Nitrogen and Water in the presence of Oxygen all at once? I.E. If the hydrazine exploded inside the Hab. Express your answer in terms of Enthalpy. How many pounds of Trinitrotoluene (TNT) would be required for the same release of Enthalpy?

As for Oxygen, the Hab has an emergency supply of 50 Liters of O2 stored in Liquid from. How many Liters of Water could he make with that if he had an unlimited supply of Hydrogen?
That’s an emergency supply, and Mark would rather keep that for, you know, emergencies. He has a better idea for getting oxygen. The Hab has a machine that is designed to convert the CO2 that the crew exhales into solid carbon, and O2 gas, so all he needs to do is release extra CO2 into the Hab atmosphere. As long as he releases it at a slower rate then the environmental regulators can pull it out and process it he should be fine.

So now he needs to figure out how to get a bunch of CO2.
That’s where the Mars Accent Vehicle (MAV) comes in. The MAV takes the crew from the surface of Mars back up to the Hermes at the end of the mission. The MAV is arguably the most critical component of a Mars mission. Because of this, the MAV for Ares 3 was landed on Mars by the previous mission. So the crew of Ares 2 landed the Ares 3 MAV on Mars four years ago. (Martinez landed the Ares 4 MAV right before the crew descended to Mars.)

Because the MAV sits around for 4 years, it manufactures it’s own fuel from the Mars atmosphere through a complex series of chemical reactions. The first step of which is to pull CO2 from the Martian atmosphere.

To save weight, when the MAV takes off, as it did with the rest of his crew on SOL 6, it leaves behind unneeded components such as the landing gear and fuel plant to save weight for the ascent. So Mark can remove the CO2 collector from the MAV fuel plant, hook it up to the HAB’s power and collect as much CO2 as he needs.

If the Hab has 92m^2 of floor space, and is at least 3 meters tall, what is the minimum internal volume of the Hab? We can get a reasonable assumption of the maximum internal volume of the Hab if we assume it is in the shape of a half sphere. What would that volume be?

If the atmospheric regulator keeps the Hab CO2 concentration at 370ppm. If CO2 becomes dangerous at 10,000ppm how much CO2 can Mark release into the Hab at once before the concentration of CO2 becomes dangerous to him based on our calculation of the minimum internal volume of the hab?

If the MAV plant can collect 10 Liters of liquid CO2 in 15 hours, how long will it take to collect enough CO2 to create enough O2 to react with his entire Hydrazine supply?

The Math of "The Martian" Sol 25

The base has a multivitamin supply that will last Mark almost 10 years, so he has enough vitamins and minerals. Careful rationing of the protein in the rations on hand will provide him with enough protein to last 4 years. So all he really needs is raw calories.

The ration packs that NASA sent with the mission were calibrated to have the proper calories per SOL. However since Mark plans to grow potatoes to supplement these ration packs, he needs to convert everything back to earth days because that’s what all his nutrition information is based on.
Ares 4, the next manned mission to Mars will land 1387 Sols from now (Sol 25), assuming it isn’t canceled due to Mark’s “death”. How many Earth days from now until Ares 4 lands?

If he eats 1500 calories per earth day his ration packs will last until SOL 392.2. How many calories does he currently have in the ration packs?

He believes that by employing aggressive farming tactics that aren’t normally necessary on earth he can grow 6.75 calories per square meter of soil per day.

He currently has 62 square meters of the Hab covered in soil. If he only uses this area to grow food, when will he run out?

The Hab has 92 square meters of floor space total. If he covers all of it with soil how long will his food last?

The two rovers also have emergency shelters that he can use, each has 10 square meters. He can also use five of the six beds to grow food, and two of the three lab tables. The bunks and lab tables are 2 square meters each.

If he uses all of those, how many square meters of farmland will he have. How long will the food last now?

How much water will he need now, for both him and the soil? How much is he short by? 

The math of "The Martian" Sol 16

Mark needs to add water to the Martian soil before he can add bacteria to it. Because the Hab has a water reclaimer each person only needs 50 liters of water to survive indefinitely. Given that, how much water did NASA send with them for the mission? If soil needs 40 Liters of water per Cubic meter, how much water will he need, including 50 liters for himself?

The Math of "The Martian" Sol 14

Scientists believe that Martian soil could grow crops if water and bacteria from earth soil is added. Mark has a small amount of earth soil that was allocated for experiments, and some live potatoes that were intended for Thanksgiving dinner. He plans to use these to turn the inside of the Habitat (Hab) into a farm. The Hab has 92 square meters of floor space, he needs to cover it with a 10 centimeter thick layer of Martian soil. What volume of soil will he need to bring into the Hab?

The Math of "The Martian" Sol 7

The mission started with enough food for 6 people for 54 Martian days (Sols). Mark plans to extend the food supply by only eating ¾ rations. If the rest of the crew left after lunch but before dinner on Sol 6, on what Sol will his food run out?

Math of "The Martian" Intro

I love the book "The Martian" mostly because unlike most entertainment these days where the hero succeeds through violence, in "The Martian" our hero succeeds through the use of math, science, and creative thinking. It such a great example of the kind of people who should be our role models, but sadly are all too often not.

I love the book so much that I am reading it for the fourth time now. But this time is different, this time, I am slowly going through, and pulling from the book as many math or science problems I can find that should be able to be solved by Junior High and High School level students.

Before we can get into the problems, I must paraphrase the first chapter of the book: 

(Note that because the length of a "day" is different on each planet, NASA uses day to refer to 24 hours, and Sol to refer to the length of a day on a given planet. Given that the story takes place on Mars, a Sol is 24 hours and 45 minutes.)

In the 2050's NASA put together the Ares missions, 5 missions to put humans on Mars. Each mission would be on the surface for roughly a month, and they would be spaced about 4 years apart. Mark Watney is the mission Botanist, and Mechanical Engineer on the 3rd Ares mission. 

On Sol 6 of their surface operations on Mars their camp is hit by a dust storm so powerful that the mission commander is forced to abort the mission early and return to earth. While the team is struggling through the dust storm to get to the rocket to leave, the main communication dish for their base camp breaks loose, it slams into the wires that make up the camp's secondary communication array. One of those wires breaks loose, hits Mark Watney and sends him flying into the storm. 

The wire doesn't kill Watney, but it does destroy the computer that relays his health data to the rest of the team, so to them it appears that he has been killed.

The remaining five members of the team try to search for him, but with visibility in the storm down to a few feet, no data coming in from Watney's health monitors, and the storm threatening to destroy the only rocket that can take them home, they are forced to leave.

On Sol 7 Mark Watney wakes to find he is alone on Mars. The next mission to Mars isn't due to touch down for 4 years, and it's slated to land almost 3200 kilometers from his camp. He has no way to contact the rest of his crew or earth, and he has supplies that were designed to last 6 people for two months...

Welcome to Matt's Math!

(formerly mattsmath233.tumblr.com)

I'm a math tutor, and I publish math and science problems and strategies that I find useful or interesting in the course of teaching.