Question: A zoologist models the number of observed behaviors per day as $ y = 2x + b $, where $ x $ is the number of hours spent in the field. If 3 hours of observation yield 11 total behaviors, find the $ y $-intercept of the line.

Title: How to Find the y-Intercept in a Linear Model: A Zoologist’s Approach

When a zoologist models observed behaviors in the wild, understanding relationships between variables is essential. One common pattern is a linear correlation, such as the equation $ y = 2x + b $, where $ y $ represents the total number of predicted behaviors per day, $ x $ is the number of observation hours per day, and $ b $ is the unknown $ y $-intercept.

In this real-world scenario, data reveals that spending 3 hours in the field results in 11 total observed behaviors. Using this information, we can determine the $ y $-intercept — the predicted number of behaviors when no time is spent observing.

We are given the linear model: $$ y = 2x + b $$ and the data point $ (x, y) = (3, 11) $. Substitute these values into the equation: $$ 11 = 2(3) + b $$ $$ 11 = 6 + b $$ Now solve for $ b $: $$ b = 11 - 6 = 5 $$

Thus, the $ y $-intercept is $ 5 $. This means that even with zero hours of observation, the model predicts 5 observed behaviors — possibly reflecting inherent activity levels or baseline activity in the environment.

In summary, for the model $ y = 2x + b $, when $ x = 3 $ and $ y = 11 $, the $ y $-intercept is $ oxed{5} $. This value helps zoologists interpret the model’s behavior, ensuring accurate and meaningful predictions from field data.