Question

rational exponents
assignment active

analyze the data using regression

graph the planet data, shown right, using the regression calculator. when you are done, click
esize window to fit data.\

	a (au)	t (yr)
venus	0.72	0.616
earth	1	1
mars	1.52	1.88
jupiter	(to be interpolated later in the activity.)
saturn
uranus	19.18	84.0
neptune	30.06	165

Explanation:

Identify the mathematical relationship in the data

Using the Rational Exponents and Radical to Exponential Form knowledge points

$$
LATEXBLOCK0
$$

Formulate the regression model

The relationship between the semi-major axis \(A\) (in astronomical units, au) and the orbital period \(T\) (in years, yr) is modeled by Kepler's Third Law:

$$ T^2 = A^3 \implies T = A^{3/2} = A^{1.5} $$

This is a power regression model of the form \(T = a \cdot A^b\), where \(a \approx 1\) and \(b \approx 1.5\).

Answer:

The data is modeled by the power regression equation:

$$T = A^{1.5}$$

or in radical form:

$$T = \sqrt{A^3}$$

where:

• \(A\) is the semi-major axis in astronomical units (au).
• \(T\) is the orbital period in years (yr).