EBK PRECALCULUS:GRAPHICAL,...-NASTA ED.
EBK PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780135795910
Author: Demana
Publisher: PEARSON CO
Expert Solution & Answer
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Chapter 5, Problem 67RE
Solution

a.

To calculate: The area of the given trapezoid as a function of θ .

The area of the given trapezoid as a function of θ is sinθ+12sin2θ .

Given information: A trapezoid is inscribed inside the upper half of a circle as shown below.

  EBK PRECALCULUS:GRAPHICAL,...-NASTA ED., Chapter 5, Problem 67RE , additional homework tip 1

Figure (1)

Formula used: The area of a trapezoid is given by Area=12sum of parallel sidesheight .

Calculation:

Draw a perpendicular to find the height of the trapezoid.

  EBK PRECALCULUS:GRAPHICAL,...-NASTA ED., Chapter 5, Problem 67RE , additional homework tip 2

Figure (2)

Apply the formula of distance between two points to calculate the length PQ .

  PQ=1+12+0−02=4=2

The radius of circle is 1 . So, RO=1 . Calculate the value of RN .

  sinθ=RNORRN=OR⋅sinθRN=1⋅sinθRN=sinθ

Calculate the value of ON .

  cosθ=ONORON=OR⋅cosθON=1⋅cosθON=cosθ

Calculate the length of RS .

  RS=2⋅ON=2cosθ

Calculate the area of trapezoid as a function of θ .

  Area=122+2cosθsinθ=221+cosθsinθ=sinθ+cosθsinθ=sinθ+122sinθcosθ

Further simplify.

  Area=sinθ+12sin2θ

Thus, the area of the given trapezoid as a function of θ is sinθ+12sin2θ .

b.

To calculate: The value of θ that maximizes the area of trapezoid. Also, find the maximum area of trapezoid.

The value of θ that maximizes the area of trapezoid is π3 and the maximum area of trapezoid is 1.299 .

Given information: A trapezoid is inscribed inside the upper half of a circle as shown below.

  EBK PRECALCULUS:GRAPHICAL,...-NASTA ED., Chapter 5, Problem 67RE , additional homework tip 3

Figure (3)

Calculation:

From part (a), the area of trapezoid is A=sinθ+12sin2θ .

Draw the graph of the area function as shown below.

  EBK PRECALCULUS:GRAPHICAL,...-NASTA ED., Chapter 5, Problem 67RE , additional homework tip 4

Figure (4)

From the graph, it can be observed that the graph reaches its maximum value for θ=π3 . So, the maximum area of the trapezoid is 1.299 .

Thus, the value of θ that maximizes the area of trapezoid is π3 and the maximum area of trapezoid is 1.299 .

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Chapter 5 Solutions

EBK PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. 5.1 - Prob. 1QRCh. 5.1 - Prob. 2QRCh. 5.1 - Prob. 3QRCh. 5.1 - Prob. 4QRCh. 5.1 - Prob. 5QRCh. 5.1 - Prob. 6QRCh. 5.1 - Prob. 7QRCh. 5.1 - Prob. 8QRCh. 5.1 - Prob. 9QRCh. 5.1 - Prob. 10QR
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Prob. 74ECh. 5.3 - Prob. 75ECh. 5.4 - Prob. 1QRCh. 5.4 - Prob. 2QRCh. 5.4 - Prob. 3QRCh. 5.4 - Prob. 4QRCh. 5.4 - Prob. 5QRCh. 5.4 - Prob. 6QRCh. 5.4 - Prob. 7QRCh. 5.4 - Prob. 8QRCh. 5.4 - Prob. 9QRCh. 5.4 - Prob. 10QRCh. 5.4 - Prob. 1ECh. 5.4 - Prob. 2ECh. 5.4 - Prob. 3ECh. 5.4 - Prob. 4ECh. 5.4 - Prob. 5ECh. 5.4 - Prob. 6ECh. 5.4 - Prob. 7ECh. 5.4 - Prob. 8ECh. 5.4 - Prob. 9ECh. 5.4 - Prob. 10ECh. 5.4 - Prob. 11ECh. 5.4 - Prob. 12ECh. 5.4 - Prob. 13ECh. 5.4 - Prob. 14ECh. 5.4 - Prob. 15ECh. 5.4 - Prob. 16ECh. 5.4 - Prob. 17ECh. 5.4 - Prob. 18ECh. 5.4 - Prob. 19ECh. 5.4 - Prob. 20ECh. 5.4 - Prob. 21ECh. 5.4 - Prob. 22ECh. 5.4 - Prob. 23ECh. 5.4 - Prob. 24ECh. 5.4 - Prob. 25ECh. 5.4 - Prob. 26ECh. 5.4 - Prob. 27ECh. 5.4 - Prob. 28ECh. 5.4 - Prob. 29ECh. 5.4 - Prob. 30ECh. 5.4 - Prob. 31ECh. 5.4 - Prob. 32ECh. 5.4 - Prob. 33ECh. 5.4 - Prob. 34ECh. 5.4 - Prob. 35ECh. 5.4 - Prob. 36ECh. 5.4 - Prob. 37ECh. 5.4 - Prob. 38ECh. 5.4 - Prob. 39ECh. 5.4 - Prob. 40ECh. 5.4 - Prob. 41ECh. 5.4 - Prob. 42ECh. 5.4 - 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