See attached images for question Free Response (from 2011 Form B AB5/BC5)Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiablefunction B models Ben's position on the track, measured in meters from the western end of thetrack, at time t, measured in seconds from the start of the ride. The table gives values for B(t)and Ben's velocity, v(t), measured in meters per second, at selected times t.• 10t (seconds)4060B(t) (meters)1001369.49m2.02.32.54.6sec A light is directly above the western end of the track. Ben rides so that at time t, the distanceL(t) between Ben and the light satisfies (L(t)) =122 +(B(t)). At what rate is the distance%3Dbetween Ben and the light changing at timet= 40?

Name: See attached images for question Free Response (from 2011 Form B AB5/B
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: See attached images for question Free Response (from 2011 Form B AB5/BC5)Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiablefunction B models Ben's position on the track, measured in meters from the western

Given: The tables gives the values for B(t) and Ben's velocity , v(t), measured in meter per second, at selected times t: t (seconds) 0 10 40 60 B(t) (meters) 100 136 9 49 v(t) msec 2.0 2.3 2.5 4.6 Now a light is directly above the western end of the tract. The distance L(t) between Ben and the light satisfies: (L(t))2=122+(B(t))2 Now we have to find that at what rate rate is the distance between Ben and the light changing at time t =40.We have, (L(t))2=122+(B(t))2 For the required result differentiate the above equation with respect to time (t) we get by the chain rule: 2 L(t) L'(t)=2 B(t) B'(t) The Chain Rule:ddt(f(g(t))=f'(g(t))×g'(t) Therefore, L'(t)=B(t) B'(t)L(t)L'(t)=B(t) v(t)L(t) B'(t)=v(t) Now we have to calculate at time (t =40) L'(40)=B(40) v(40)L(40) Now from the given table: L'(40)=9×2.5122+92 =22.5144+81 =22.5225 =22.515 =1.5 m/sec Hence, at 1.5 m /sec rate is the distance between Ben and the light changing at time t =40.