## Problem Set

# Flying Formula: A 'Pi in the Sky' Math Challenge

### Overview

The "Pi in the Sky" math challenge gives students a chance to take part in recent discoveries and upcoming celestial events, all while using math and pi just like NASA scientists and engineers. In this problem from the set, students use the mathematical constant pi to calculate how much fuel the Cassini spacecraft consumed after years of orbiting Saturn.

### Materials

### Background

Cassini began the first in-depth, up-close study of Saturn and its system of rings and moons in 2004. It became the first spacecraft to orbit Saturn, beginning a mission that yielded troves of new insights over more than a decade. The Saturnian system proved to be rich ground for exploration and discoveries, and Cassini's science findings changed the course of future planetary exploration.

The Cassini mission had two mission extensions, allowing for more flybys, investigations and measurements, over a longer span of time.

When its initial four-year tour of the Saturn system was complete in 2008, the Cassini-Huygens saga had brought a new dimension of understanding to the complex and diverse Saturn system.

The two-year Cassini Equinox Mission brought continued excitement. During that first extended mission, the spacecraft made 60 additional orbits of Saturn, 26 flybys of Titan, seven of Enceladus, and one each of Dione, Rhea and Helene. The Equinox mission allowed for observations of Saturn's rings as the sun lit them edge-on, revealing a host of never-before-seen insights into the rings' structure.

In 2010, the spacecraft began a second, seven-year-long, extended mission called the Cassini Solstice Mission. This final mission concluded with a phase known as The Grand Finale -- 22 deep dives between Saturn's cloud tops and innermost ring before it plunged into the giant planet's atmosphere.

### Procedures

- The Cassini spacecraft was launched to Saturn with its 28-inch spherical hydrazine tank filled to 69 percent of its volume with hydrazine. After many years of studying Saturn, 82 kilograms of hydrazine were used to maneuver around the ringed planet. Given the density of hydrazine is 1.02 grams/cubic centimeter, how much fuel remained in the tank at this time? (Assume no fuel is sitting in the fuel lines.)

### Assessment

### Extensions

Pi Day Challenges- Pi in the Sky
- Pi in the Sky 2
- Pi in the Sky 3
- Pi in the Sky 4
- Pi in the Sky Challenge (slideshow for students)

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