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Modeling Power Growth in Astronomy: Insights from the Xena Discovery

In this section, we explore power law models applicable to astronomical data. Power growth can be represented by a specific equation, where constants a and b signify growth rates. By utilizing natural logarithms, we demonstrate how a linear relationship emerges between ln(y) and ln(x), correlating to data on celestial bodies. We analyze the discovery of a new planetary body, Xena, which exceeds Pluto in size. Through scatterplots and regression analysis, we assess the relationship between distance and period, ultimately predicting Xena's orbital period.

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Modeling Power Growth in Astronomy: Insights from the Xena Discovery

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  1. AP Statistics: Section 4.1 CPower Law Models

  2. Power growth can be modeled by the equation _______, where a and b are constants. (Notice that in exponential growth, x is the _________, while in power growth, x is the _____.) exponent base

  3. We can model power growth using a LSL as follows:Take the ln of both sides.The ln of a product equals the sum of the ln’s.The ln of a power equals the power times the ln.

  4. Notice that there is a linear relationship between lnyand lnxand that b is the slope of the straight line that links lny to lnx.

  5. On July 31, 2005, a team of astronomers announced they had discovered what seemed to be a new planet in our solar system. Xena, the potential planet, is bigger than Pluto and has an average distance from the sun of 9.5 billion miles (102.15 astronomical units). Could Xena be a new planet?

  6. Graph the scatterplot of the planetary data. Describe it. Positive Curved Strong

  7. Graph distance vs. ln (period). Describe it. Positive Curved Strong

  8. Graph ln (distance) vs. ln (period). Describe it. Positive Linear Strong

  9. r2 = ________ and the LSL is ____________________________

  10. Plot the residuals. Discuss. There is a curved pattern, BUT the residuals are really quite small so we will live with it.

  11. Find the prediction equation and predict the period for Xena.

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