Riemann sums and integrals 1060(seconds)B(t)(meters)100 136 9492.0 2.32.5 4.6(meters per second)5. Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function B modelsBen's position on the track, measured in meters from the western end of the track, at time t, measured inseconds from the start of the ride. The table above gives values for B(t) and Ben's velocity, v(t), measured inmeters per second, at selected times t.(a) Use the data in the table to approximate Ben's acceleration at timet = 5 seconds. Indicate units of measure.(b) Using correct units, interpret the meaning of v(t) dt in the context of this problem. Approximate-60|v(t) dt using a left Riemann sum with the subintervals indicated by the data in the table.(c) For 40 sIS 60, must there be a time t when Ben's velocity is 2 meters per second? Justify your answer.(d) A light is directly above the western end of the track. Ben rides so that at time t, the distance L(t) betweenBen and the light satisfies (L(t)) = 122 + (B(t)). At what rate is the distance between Ben and the light%3Dchanging at time t = 40 ?40

Answered: (seconds) 10 40 60 B(t) (meters) 100…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Riemann sums and integrals

10
60
(seconds)
B(t)
(meters)
100 136 9
49
2.0 2.3
2.5 4.6
(meters per second)
5. Ben rides a unicycle back and forth along a straight east-west track. The twice-differentiable function B models
Ben's position on the track, measured in meters from the western end of the track, at time t, measured in
seconds from the start of the ride. The table above gives values for B(t) and Ben's velocity, v(t), measured in
meters per second, at selected times t.
(a) Use the data in the table to approximate Ben's acceleration at timet = 5 seconds. Indicate units of measure.
(b) Using correct units, interpret the meaning of v(t) dt in the context of this problem. Approximate
-60
|v(t) dt using a left Riemann sum with the subintervals indicated by the data in the table.
(c) For 40 sIS 60, must there be a time t when Ben's velocity is 2 meters per second? Justify your answer.
(d) A light is directly above the western end of the track. Ben rides so that at time t, the distance L(t) between
Ben and the light satisfies (L(t)) = 122 + (B(t)). At what rate is the distance between Ben and the light
%3D
changing at time t = 40 ?
40

Expert Solution

Step 1

Ben's Acceleration at time $t = 5 s e c o n d s$

from the table we can see that t $= 5 s e c$ is the average of $t = 10 s e c$ and $t = 0 s e c$

So from here it is clear that question is about average acceleration

so we are

$a v g A c c e l a r a t i o n = \frac{\vec{v_{2}} (t_{2}) - \vec{v_{1}} (t_{1})}{t_{2} - t_{1}}$

Here

$t_{2} = 10 s e c t_{1} = 0 s e c v_{2} (t_{2}) = 2.3 m e t e r p e r s e c o n d (m / s) v_{1} (t_{1}) = 2.0 (m / s) ∴ A v g . A c c e l a r a t i o n = \frac{2.3 m / s - 2.0 m / s}{10 s e c - 0 s e c} = \frac{(0.3) m / s}{10 s e c} a v g A c c e l e r a t i o n = 0.03 m / s^{2}$

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