# National Aeronautics and Space Administration

## Year 5: Problems 149 to 234

Problem 234: Calculating Arc Lengths of Simple Functions- Students work with the differential form of the Pythagorean Theorem to determine the basic integral formula for arc length, then evaluate it for a parabola, logrithmic spiral and normal spiral. They evaluate the length of the spiral track on a CDrom. [Grade: 11-12 | Topics: Calculus; differential; integral, U-substitutions; significant figures.]

Problem 233: The Milky Way: A mere cloud in the cosmos- Students compare the average density of the Milky Way with the density of the universe. [Grade: 8-10 | Topics: Volume of disk, density, scientific notation]

Problem 232: Star Circles- Students use a photograph of star trails around the North Star Polaris to determine the duration of the timed exposure based on star arc lengths. [Grade: 8-9 | Topics: Lengths of arcs of circles; angular measure.]

Problem 231: Star Magnitudes and Decimals- Students work with the stellar magnitude scale to determine the brightness differences between stars. [Grade: 5-8 | Topics: Multiplying decimals.]

Problem 230: Galaxy Distances and Mixed Fractions- Students use the relative distances to nearby galaxies expressed in mixed numbers to determine distances between selected galaxies. [Grade: 3-5 | Topics: Basic fraction math.]

Problem 229: Atomic Numbers and Multiplying Fractions- Students use a piece of the Periodic Table of the Elements to figure out the identities of atoms based on numerical clues expressed as mixed numbers. [Grade: 3-5 | Topics: Basic fraction math; mixed numbers.]

Problem 228: Nuclear Arithmetic- Students use the equation N = A - Z to solve for A, Z or N given values for the other two variables. [Grade: 4-6 | Topics: Evaluating a simple equation.]

Problem 227: Working With Rates- Students examine mixed rates for a variety of situations and their connections to ratios. [Grade: 6-8 | Topics: Ratios; scientific notation; unit conversion.]

Problem 226: Rates and Slopes: An astronomical perspective- Students determine the slopes for two linear graphs and make the connection to rates with mixed units. [Grade: 7-9 | Topics: Finding the slope of a linear graph.]

Problem 225: Areas Under Curves; An astronomical perspective- Students work with a bar graph of the number of planet discoveries since 1995 to evaluate the total discoveries, as areas under the graph, for various combinations of time periods. [Grade: 6-8 | Topics: Adding areas in bar graphs.]

Problem 224: Perimeters; Which constellation is the longest?- Students use tabulated data for the angular distances between stars in the Big Dipper and Orion to determine which constellation has the longest perimeter, and the average star separations. [Grade: 3-5 | Topics: perimeter of a curve; basic fractions; mixed numbers.]

Problem 223: Volcanos are a Blast: Working with simple equations- Students examine the famous Krakatoa explosion, asteroid impacts on the moon, and geysers on Enceladus using three equations that describe the height of the plume and initial velocity, to answer questions about the speed of the debris and terminal height. [Grade: 9-11 | Topics: Algebra I; significant figures.]

Problem 222: Kelvin Temperatures and Very Cold Things- Students convert from Centigrade to Fahrenheit and to Kelvin using three linear equations. [Grade: 5-8 | Topics: Evaluating simple linear equations for given values.]

Problem 221: Pulsars and Simple Equations- Students work with linear equations describing the rotation period of a pulsar, and evaluate the equations for various conditions. Students use the equations to predict intersection points in time. [Grade: 6-8 | Topics: Evaluating simple one-variable equations]

Problem 220: The Many Faces of Energy- Students convert between several different energy units. [Grade: 8-10 | Topics: Scientific notation; unit conversions.]

Problem 219: Variables and Expressions from Around the Cosmos- Students evaluate linear equations describing a variety of astronomical situations. [Grade: 6-8 | Topics: Evaluating simple one-variable equations.]

Problem 218: Craters are a Blast! - Students measure crater diameters in a photo of the moon, and determine the energy required to create them using a simple quadratic equation. [Grade: 8-10 | Topics: Scientific notation; evaluating simple power equations.]

Problem 217: Fractions and Chemistry- Students study simple chemical equations by using simple proportions and mixed numbers. [Grade: 3-6 | Topics: Basic fraction math; ratios.]

Problem 216: Atomic Fractions- Students study the energy ladders of an atom and work out, using differences between mixed numbers, the energy gained or lost by an electron as it moves up and down the ladder. [Grade: 3-6 | Topics: Basic fraction math]

Problem 215: More Atomic Fractions- Students study the energy ladders of an atom and work out, using differences between mixed numbers, the energy gained or lost by an electron as it moves up and down the ladder. [Grade: 3-6 | Topics: Basic fraction math.]

Problem 214: Atomic Fractions III- Students study the energy ladders of an atom and work out, using differences between mixed numbers, the energy gained or lost by an electron as it moves up and down the ladder. [Grade: 3-6 | Topics: Basic fraction math.]

Problem 213: Kepler: The hunt for Earth-like planets- Students compare the area of a star with the area of a planet to determine how the star's light is dimmed when the planet passes across the star as viewed from Earth. This is the basis for the 'transit' method used by NASA's Kepler satellite to detect new planets. [Grade: 6-8 | Topics: Area of circle; ratios; percents.]

Problem 212: Finding Mass in the Cosmos- Students derive a simple formula, then use it to determine the masses of objects in the universe from the orbit periods and distances of their satellites. [Grade: 9-12| Topics: Scientific Notation; Algebra II; parametric equations]

Problem 211: Where Did All the Stars Go?- Students learn why NASA photos often don't show stars because of the way that cameras take pictures of bright and faint objects. [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 210: The Mathematics of Ion Rocket Engines- Students learn about the basic physics of ion engines, calculating speeds. [Grade: 9-12| Topics: Scientific Notation; Algebra II; evaluating formulae.]

Problem 209: How to make faint things stand out in a bright world!- Students learn that adding images together often enhances faint things not seen in only one image; the power of averaging data. [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 208: Optimization- Students determine the optimal dimensions of an hexagonal satellite to maximize its surface area given its desired volume. [Grade: 9-12| Topics: Calculus; differentiation.]

Problem 207: The STEREO Mission: getting the message across- Students learn about how the transmission of data is affected by how far away a satellite is for the two satellites in the STEREO constellation. [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 206: Can You Hear me now? - Students learn about how the transmission of data is affected by how far away a satellite is, for a variety of spacecraft in the solar system [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 205: The Io Plasma Torus- Students approximate the Io radiation belts as a cylinder to determine its volume ,and the mass of the particles within it. [Grade: 9-12| Topics: Algebra I - volume of cylinders; calculus - Integrals of volumes.]

Problem 204: The Mass of the Van Allen Radiation Belts- Students graph some magnetic field lines in polar coordinates, then estimate the volume and mass of the Belts using the formula for a torus. [Grade: 9-12| Topics: Algebra II.]

Problem 203: Light Travel Times- Students determine the time it takes light to reach various objects in space. [Grade: 6-8| Topics: Scientific Notation; Multiplication; time = distance/speed.]

Problem 202: The Dawn Mission - Ion Rockets and Spiral Orbits- Students determine the shape of the trajectory taken by a spacecraft using a constant-thrust ion motor using differential and integral calculus for arc lengths. [Grade: 9-12| Topics: Calculus - Arc lengths.]

Problem 201: Fly Me To the Moon!- Students learn some basic principles and terminology about how spacecraft change their orbits to get to the moon. [Grade: 6-8| Topics: speed = distance/time; Pythagorean Theorem]

Problem 200: The Moon's Density - What's Inside?- Students develop a simple mathematical model of the moon's interior using two nested spheres with different densities. [Grade: 9-12| Topics: Volume of a sphere; mass = density x volume.]

Problem 199: Solar Storm Energy and Pie Graphs- Students study two Pie graphs describing solar flares and draw conclusions about percentages and their various forms of energy. [Grade: 6-8| Topics: Interpreting Pie Graphs.]

Problem 198: Solar Storm Timeline- Students read a narrative about the events involved in a solar storm, create a chronology for the sequence of events, and answer some simple time-related questions. [Grade: 6-8| Topics: Time calculations.]

Problem 197: Hubble Sees a Distant Planet- Students study an image of the dust disk around the star Fomalhaut and determine the orbit period and distance of a newly-discovered planet orbiting this young star. [Grade: 6-10| Topics: Calculating image scales; Circle circumferences; Unit conversions; distance-speed-time]

Problem 196: Angular Size and velocity- Students study a spectacular photo of the ISS passing across the face of the sun, and work out the angular sizes and speeds of the transit to figure out how long the event took in order to photograph it. [Grade: 8-10| Topics: Geometry; Angle measurement]

Problem 195: Unit Conversions III- Students work with more unit conversions and use them to solve a series of practical problems in science and solar energy. [Grade: 6-10| Topics: unit conversions.]

Problem 194: A Magnetic Case for 'What Came First?' - Students create a timeline for events based on several data plots from the THEMIS program, and use their timeline to answer questions about the causes of magnetic storms. [Grade: 6-8| Topics: Time calculations]

Problem 193: Fluid Level in a Spherical Tank - Students explore the relationship between volume, and the height of fluid in a spherical tank as fluid is being drained at a constant rate. [Grade: 10-12| Topics: Algebra, differential calculus, related rates]

Problem 192: The Big Bang - Cosmic Expansion - Students explore the expansion of the universe predicted by Big Bang cosmology [Grade: 10-12| Topics: Algebra, Integral Calculus]

Problem 191: Why are hot things red? - Students explore the Planck Function using graphing skills, and calculus for experts, to determine the relationship between temperature and peak wavelength. [Grade: 10-12| Topics: Algebra, graphing, differential calculus]

Problem 190: Modeling a Planetary Nebula - Students use calculus to create a mathematical model of a planetary nebula [Grade: 10-12| Topics: Algebra, Integral calculus]

Problem 189: Stellar Temperature, Size and Power- Students work with a basic equation to explore the relationship between temperature, surface area and power for a selection of stars. [Grade: 8-10| Topics: Algebra]

Problem 188: Cross Sections and Collision Times - Students explore the relationship between density, speed and size in determining how quickly particles collide in a gas. [Grade: 9-11| Topics: Algebra; Area]

Problem 187: Differentiation- Students explore partial derivatives by calculating rates of change in simple equations taken from astrophysics. [Grade: 11-12| Topics: differentiation; algebra]

Problem 186: Collapsing Gas Clouds and Stability- Students use the derivative to find an extremum of an equation governing the pressure balance of an interstellar cloud. [Grade: 11-12| Topics: differentiation; finding extrema; partial derivitives]

Problem 185: The International Space Station: Follow that graph!- Students use a plot of the orbit altitude of the ISS to predict its re-entry year after the peak of the next solar activity cycle. [Grade: 6-8| Topics: extrapolating a simple graph; estimation; forecasting]

Problem 184: The Ant and the Turntable: Frames of reference - Students predict the motion of an ant crawling from the center of a spinning CDrom to the edge. They also use calculus to estimate the length of the spiral path seen by a stationary observer. [Grade: 11-12| Topics: integration; parametric equations; polar coordinates]

Problem 183: Calculating Arclengths of Simple Functions- Students determine the basic equation for arclength and its integral, and evaluate it for simple polar functions. [Grade: 11-12| Topics: calculus; integration; parametric equations]

Problem 182: Our Neighborhood in the Milky Way- Students create a scale model of the local Milky Way and estimate distances and travel times for a series of voyages. [Grade: 6-8| Topics: scale models; speed-distance-time]

Problem 181: Extracting Oxygen from Moon Rocks- Students use a chemical equation to estimate how much oxygen can be liberated from a sample of lunar soil. [Grade: 9-11| Topics: ratios; scientific notation; unit conversions]

Problem 180: Planets, Fractions and Scales- Students work with relative planet comparisons to determine the actual sizes of the planets given the diameter of Earth. [Grade: 4-6| Topics: scale models; decimals; fractions]

Problem 179: Is There a Lunar Meteorite Impact Hazard? - Students work with areas, probability and impact rates to estimate whether lunar colonists are in danger of meteorite hazards. [Grade: 5-7| Topics: Area; unit conversions; rates]

Problem 178: The Mass of the Moon - Students use the period and altitude of a NASA lunar spacecraft to determine the mass of the moon. [Grade: 8-11| Topics: Algebra]

Problem 177: Lunar Cratering: Probability and Odds- Students work with crater counting to estimate the area covered by craters and how to convert this into impact probabilities. [Grade: 4-7| Topics: Area; probability]

Problem 176: Solar Storms: Sequences and Probabilities I - Students continue their study of a stormy week on the sun by working out the probabilities for joint events. [Grade: 4-7| Topics: probability; numerating possible outcomes]

Problem 175: Solar Storms: Sequences and Probabilities II - Students work out the probabilities for various combinations of solar storms during a given week. [Grade: 4-7| Topics: probability; numerating possible outcomes]

Problem 174: A Matter of Timing - Students study satellites of Saturn to work out graphically how often they will line up. [Grade: 3-6| Topics: scale model; time calculation; fractions; working with rulers and compasses]

Problem 173: Groups, Clusters and Individuals- Students determine the number of individual objects given the number of groups and the number of individuals in an average group for clusters of stars and galaxies. [Grade: 3-5 | Topics: multiplication]

Problem 172: The Stellar Magnitude Scale- Students learn about positive and negative numbers using a popular brightness scale used by astronomers. [Grade: 3-6| Topics: number relationships; decimals; negative and positive numbers]

Problem 171: More Unit Conversions- Students work with four unit conversion problems that are a bit tricky! [Grade: 6-8 | Topics: unit conversions]

Problem 170: Measuring Star Temperatures- Students use a simple formula to determine the temperatures of stars, and to use a template curve to analyze data for a specific star to estimate its temperature. [Grade: 6-8 | Topics: algebra, graph analysis]

Problem 169: The Limiting Behavior of Functions- Students work with two complex formulae to determine their limiting behavior as the independent variables approach infinity and zero. [Grade: 9-12 | Topics: Algebra II, pre-calculus]

Problem 168: Fitting Periodic Functions - Distant Planets- Students work with data from a newly-discovered extra-solar planet to determine its orbit period and other parameters of a mathematical model. [Grade: 9-12 | Topics: trigonometry; functions; algebra]

Problem 167: Areas and Probabilities- Students work with various means of estimating probabilities using a comparison of areas. [Grade: 4-6 | Topics: percentages,calculating areas; probability]

Problem 166: The Dollars and Cents of Research - Students work with dollar amounts, hourly salary rates, percentages to explore various models of the cost of scientific research as seen by the individual scientist. [Grade: 4-6 | Topics: percentages, decimal math, simple rates (e.g dollars/hour)]

Problem 165: Fractions in Space - Students explore the many ways that simple fractions come up in the study of planetary motion. [Grade: 3-5 | Topics: working with fractions; time calculations]

Problem 164: Equations with One Variable- Students work with equations like '4.3 = 3.26D' to solve for D in a number of simple astronomical problems involving distances, speed and temperature conversion. [Grade: 6-8 | Topics: equations in one variable; multiplication; division; decimals]

Problem 163: Time Intervals - Students calculate time intervals between a number of astronomical events, from milliseconds to years. [Grade: 3-5 | Topics: Time calculations; unit conversions; decimal math]

Problem 162: Areas and Probabilities- Students explore how area and probability are related in working out meteor impact and 'stormy day' problems. [Grade: 4-6 | Topics: Working out all possible outcomes for a set of events; fractions and percentages]

Problem 161: Earth and Moon to Scale- Students create a scale model of the Earth-Moon system and compare with artistic renditions and actual NASA spacecraft images. [Grade: 4-6| Topics: Decimals; scaling and similarity]

Problem 160: The Relative Sizes of the Sun and Stars- Students work through a series of comparisons of the relative sizes of the sun compared to other stars, to create a scale model of stellar sizes using simple fractional relationships. ( e.g if Star A is 6 times larger than Star B, and Star C is 1/2 the size of Star B, how big is Star C in terms of Star A?) [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 159: Galaxies to Scale - Students explore the relative sizes of the Milky Way compared to other galaxies to create a scale model of galaxies, similar to the methods in Problem 161. [Grade: 4-6 | Topics: working with fractions; scale models]

Problem 158: The Solar Neighborhood within 17 Light Years - Students create a scale model of the local solar neighborhood and determine the shortest travel distances to several stars. [Grade: 6-8 | Topics: Plotting polar coordinates using a ruler and compass; decimal math]

Problem 157: Space Shuttle Launch Trajectory - I - Students use the parametric equation for the altitude and range for an actual Shuttle launch to determine the speed and acceleration of the Shuttle during launch and orbit insertion. [Grade: 11-12 | Topics: Algebra; Calculus; Parametric Equations; Differentiation

Problem 156: Spectral Classification of Stars- Students use actual star spectra to classify them into specific spectral types according to a standard ruberic. [Grade: 5-8 | Topics: Working with patterns in data; simple sorting logic

Problem 155: Tidal Forces: Let 'er rip! - Students explore tidal forces and how satellites are destroyed by coming too close to their planet. [Grade: 7-10| Topics: Algebra; number substitution]

Problem 154: Pan's Highway and Saturn's Rings - Students use an image from the Cassini spacecraft to determine how large the satellite Pan is, and the scale of Saturn's rings using a millimeter ruler. [Grade: 4-6 | Topics:Finding the scale of an image; measurement; unit conversion]

Problem 153: Number Sentence Puzzles - Students are presented with three number sentences such as 145 + N = 375, and asked to select which 'spacy' word problem they belong to. [Grade: 3-4 | Topics: Number Topics and problem solving]

Problem 152: The Hubble Law - Students plot the speed and distance to 7 galaxies and by deriving the slop of the linear model for the data points, obtain an estimate for the expansion rate of the universe known as Hubble's Constant. [Grade: 6-8 | Topics: Plotting data; determining the slope of the data;]

Problem 151: Time Zone Math - Students learn about time zones and perform simple clock calculations using common United States and European time zones. [Grade: 3-5 | Topics: time units; addition, subtraction]

Problem 150: Cosmic Bar Graphs - Students interpret simple bar graphs taken from astronomical data. [Grade: 3-5 | Topics: finding maxima and minima; fractions; extrapolating data]

Problem 149: Equations with One Variable - Students solve formulas of the form 2001 = 1858 + 11x to find 'X'. [Grade: 3-5 | Topics: addition, subtraction, multiplication, division; solving simple equations]