Streaming services often use artificial intelligence to recommend music. One way to do this is to represent song attributes and a listener’s preferences using numbers or points on a grid. Recommendations for a listener can then be made by looking for songs represented by points that are close in distance to the point representing the listener’s preferences.
A particular music app represents song attributes and listener preferences using two scores: Energy and Happiness, which are each integers from \(1\) to \(10\). Suppose that three songs are represented by the following points:
| Song | (Energy, Happiness) |
|---|---|
| \(A\) | \((5,5)\) |
| \(B\) | \((4,3)\) |
| \(C\) | \((2,8)\) |
Denise’s song preference score is represented as \((6,4)\). Use the distance formula \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) to calculate the distance from Denise’s song preference score to each song. Which of the three songs would the streaming service recommend to Denise?
After an app update, a song is recommended if its distance from the listener’s score is most \(4\). How many different scores of \((E,H)\) could there be in the songs recommended to Denise?