Question

A music streaming app uses the function f(x) = `tan^-1 (x/10)` to assign a mood score based on the number of hours a user listens to music per week. Let the listening times (in hours/week) of two users, User A and User B, be 6 and 8 respectively. Compute the combined mood score of user A and user B, that is, f(6) + f(8).

Answer 1

Given, f(x) = `tan^-1 (x/10)`

For user A, who listens to music for 6 hours per week, the mood score is:

f(6) = `tan^-1 (6/10)`

For user B, who listens for 8 hours per week, the mood score is:

f(8) = `tan^-1 (8/10)`

Therefore, the combined mood score is:

f(6) + f(8)

= `tan^-1 (6/10) + tan^-1 (8/10)`

To simplify this, we use the identity tan⁻¹ a + tan⁻¹ b = `tan^-1((a + b)/(1 - ab)​)`, when ab < 1.

Here, a = `6/10`, b = `8/10`

ab = `48/100 < 1`

so the identity is applicable.

Applying it, we get

f(6) + f(8)

= `tan^-1 ((6/10 + 8/10)/(1 - 6/10 xx 8/10))`

= `tan^-1 ((14/10)/(1 - 48/100))`

= `tan^-1 ((14/10)/(100/100 - 48/100))`

= `tan^-1 ((14/10)/(52/100))`

= `tan^-1 (((7 xx 2)/(5 xx 2))/((13 xx 4)/(25 xx 4)))`

= `tan^-1 ((7/5)/(13/25))`

= `tan^-1  7/5 xx 25/13`

= `tan^-1  (7 xx 5)/13`

= `tan^-1 (35/13)`