Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 6 - Applications of Integration - 6.1 Velocity and Net Change - 6.1 Exercises - Page 408: 27

Answer

$$s\left( t \right) = 10 + 70t - 16{t^2}{\text{ and }}v\left( t \right) = 70 - 32t$$

Work Step by Step

$$\eqalign{ & a\left( t \right) = - 32,\,\,\,\,v\left( 0 \right) = 70,\,\,\,\,s\left( 0 \right) = 10 \cr & \cr & {\text{use theorem 6}}{\text{.2 }} \cr & {\text{Given the acceleration }}a\left( t \right){\text{ of an object moving along a line and its initial velocity }}v\left( 0 \right){\text{, }} \cr & {\text{the velocity of the object for future time }}t \geqslant 0{\text{ is }}v\left( t \right) = v\left( 0 \right) + \int_0^t {a\left( x \right)} dx \cr & {\text{let }}a\left( x \right) = - 32{\text{ and }}v\left( 0 \right) = 70.{\text{ then }}v\left( t \right){\text{ is}} \cr & v\left( t \right) = 70 + \int_0^t {\left( { - 32} \right)} dx \cr & {\text{integrate}} \cr & v\left( t \right) = 70 + \left( { - 32x} \right)_0^t \cr & {\text{evaluate the limits}} \cr & v\left( t \right) = 70 + \left( { - 32\left( t \right) - 32\left( 0 \right)} \right) \cr & {\text{simplify}} \cr & v\left( t \right) = 70 - 32t \cr & \cr & {\text{use theorem 6}}{\text{.1 }} \cr & {\text{Given the velocity }}v\left( t \right){\text{ of an object moving along a line and its initial position }}s\left( 0 \right){\text{, }} \cr & {\text{the position function of the object for future time }}t \geqslant 0{\text{ is }}s\left( t \right) = s\left( 0 \right) + \int_0^t {v\left( x \right)} dx \cr & {\text{let }}v\left( x \right) = 70 - 32x{\text{ and }}s\left( 0 \right) = 10.{\text{ then }}s\left( t \right){\text{ is}} \cr & s\left( t \right) = 10 + \int_0^t {\left( {70 - 32x} \right)} dx \cr & {\text{integrate}} \cr & s\left( t \right) = 10 + \left( {70x - 16{x^2}} \right)_0^t \cr & {\text{evaluate the limits}} \cr & s\left( t \right) = 10 + \left( {70\left( t \right) - 16{{\left( t \right)}^2}} \right) - \left( {70\left( 0 \right) - 16{{\left( 0 \right)}^2}} \right) \cr & {\text{simplify}} \cr & s\left( t \right) = 10 + 70t - 16{t^2} \cr} $$
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