Question

how is the distance from the sun for planets in our solar system related to the mean temperature of each planet? to find out, scatterplots that relate the distance of each planet (including pluto) from the sun in millions of miles and the mean planetary temperature in kelvin were created. some of the scatterplots use the natural log to transform one, or both, of the variables. based on the scatterplots, what type of model is most appropriate for comparing distance and temperature? an exponential model is appropriate because the scatterplot relating distance to temperature is curved. a linear model is appropriate because the scatterplot relating ln(distance) to ln(temperature) is roughly linear. a power model is appropriate because the scatterplot relating ln(distance) to ln(temperature) is roughly linear. an exponential model is appropriate because the scatterplot relating distance to ln(temperature) is roughly linear.

Explanation:

A power model has the form \( y = ax^b \). Taking the natural - log of both sides gives \( \ln(y)=\ln(a)+b\ln(x) \). If the scatter - plot of \( \ln(x) \) versus \( \ln(y) \) is roughly linear, then a power model is appropriate. An exponential model has the form \( y = ae^{bx} \), and taking the natural - log gives \( \ln(y)=\ln(a)+bx \). A linear model has the form \( y = ax + b \).
The key is to recall the transformation rules for different models. For a power model \( y = kx^{n} \), when we take the natural - log of both variables (\( \ln(y)\) and \( \ln(x)\)), the relationship becomes \( \ln(y)=\ln(k)+n\ln(x) \), which is a linear relationship in terms of \( \ln(x) \) and \( \ln(y) \).

Answer:

A power model is appropriate because the scatterplot relating \( \ln(\text{distance}) \) to \( \ln(\text{temperature}) \) is roughly linear.