BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.6, Problem 13E
To determine

To find: The velocity of the ball when t=2.

Expert Solution

Answer to Problem 13E

The velocity of the ball when t=2 is 24 ft/s_.

Explanation of Solution

Given:

The height (in feet) of the ball after t seconds is y=40t16t2.

The position function is, y=40t16t2.

Formula used:

The derivative f(a) is the velocity v(t) of the particle at time t=a.

v(t)=limtaf(t)f(a)ta (1)

Calculation:

Obtain the velocity of the ball when time t=2.

Take the position function y=f(t) and substitute time a=2 in equation (1)

v(t)=limt2f(t)f(2)t2=limt2(40t16t2)(40(2)16(2)2)t2=limt2(40t16t2)(8064)t2=limt240t16t216t2

=limt2(16t240t+16)t2=limt28(2t25t+2)t2

Factorize the numerator,

v(t)=limt28(2t24tt+2)(t2)=limt28(2t(t2)1(t2))(t2)=limt28(2t(t2)1(t2))(t2)=limt28(2t1)(t2)(t2)

Cancel the common term (t2) from both the numerator and the denominator,

v(t)=limt2(8(2t1))=8limt2(2t1)

Simplify further and obtain the velocity as follows.

v(t)=8(2(2)1)=8(41)=8(3)=24

Thus, the velocity of the ball at time t=2 is 24 ft/s.

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