Answered: The function s=t3−12t2+48t, 0≤t≤5,…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.7: More On Inequalities

Problem 44E

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Question

The function

s=t3−12t2+48t,

0≤t≤5,

gives the position of a body moving on a coordinate line, with s in meters and t in seconds.

a. Find the body's displacement and average velocity for the given time interval.

b. Find the body's speed and acceleration at the endpoints of the interval.

c. When, if ever, during the interval does the body change direction?

Expert Solution

Step 1

$s (t) = t^{3} - 12 t^{2} + 48 t 0 \leq t \leq s a . N o w d i s p l a c e m e n t △ d = s (s) - s (0) = - 12 * 5^{2} + 48 * S - 0 + 0 = - 60 u n i t a v e r a g e v e l o c i t y V_{a r g} = \frac{△ d}{△ t} = \frac{- 60}{5 - 0} = - 12 u n i t / t i m e b . d i f f e r e n t i a t i n g s (t) = - 12 t^{2} + 48 t + 48 t w e g e t v (t) = \frac{d s}{d t} = - 24 t + 48 u n i t / t i m e a (t) = \frac{d^{2} s}{d t^{2}} = - 24 u n i t / t i m e s o a t t = 0, v (0) = 48 u n i t / t i m e a (0) = - 24 u n i t / t i m e^{2} a t t = 5 | v (5) | = | - 24 * 5 + 48 | = 72 u n i t / t i m e a (5) = - 24 u n i t / t i m e c . T h e d i r e c t i o n o f t h e b o d y d e t e r m i n e d b y i t s v e l o c i t y . w h e n b o d y m o v i n g f o r w a r d t h e n v e l o c i t y i s p o s i t i v e a n d f o r m o v i n g b a c k w a r d v e l o c i t i s n e g a t i v e$

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