Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Question

Chapter 2.1, Problem 3E

To determine

To find: The speed of the object at t=3 seconds. Confirm the answer algebraically.

Expert Solution & Answer

Answer to Problem 3E

The speed of the object at t=3 seconds is 96 ft/sec .

Explanation of Solution

Given information:

An object is dropped from rest from the top of a building falls y=16t2 feet in first t seconds.

Calculation:

The instantaneous speed of an object at t=a seconds is the average speed from time t=a seconds to t=a+h seconds.

Use the formula for average speed.

s=ΔyΔt=y2−y1t2−t1

Substitute 3+h for t in the given equation for the distance travelled.

y2=16(3+h)2

Substitute 3 for t in the given equation for the distance travelled.

y1=16(3)2=16×9=144

Substitute these values in the formula for average speed.

s=y2−y1t2−t1=16(3+h)2−1443+h−3=16(3+h)2−144h

Substitute 0.0001 for h in the above expression for approximate speed.

s=16(3+h)2−144h=16(3+0.0001)2−1440.0001=16(3.0001)2−1440.0001=144.0096−1440.0001

Further simplify.

s=144.0096−1440.0001=0.00960.0001=96 ft/sec

So, the instantaneous speed of the object at t=3 seconds is approximately 96 ft/sec .

Verify the result algebraically:

Use the formula for speed of the object algebraically.

s=y2−y1t2−t1=16(3+h)2−1443+h−3=16(3+h)2−144h=16(9+h2+6h)−144h

Further, simplify.

s=16(9+h2+6h)−144h=144+16h2+96h−144h=h(16h+96)h=16h+96

Substitute 0 for h for the exact instantaneous speed.

s=16h+96=(16×0)+96=96 ft/sec

Therefore, the speed of the object at t=3 seconds is 96 ft/sec .

Chapter 2.1, Problem 2E

Chapter 2.1, Problem 4E

Chapter 2 Solutions

Calculus: Graphical, Numerical, Algebraic

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The speed of the object at t = 3 seconds. Confirm the answer algebraically.