Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Question

Chapter 2.1, Problem 63E

To determine

To draw:The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end.

Expert Solution

Explanation of Solution

Given:

The perimeter of the room is 200-meter single-lane running track. Length and width of the rectangular region are denoted by x and y respectively.

Formula/ concept used:

A rectangle is four sided figure each angle 900 .

Drawing:

The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end is shown in Figue-1 here.

Precalculus with Limits: A Graphing Approach, Chapter 2.1, Problem 63E , additional homework tip 1

To determine

The radius of semicircular ends, and the distance in terms of y , around the inside edge of the two semicircular parts of the track.

Expert Solution

Answer to Problem 63E

The radius of semicircular ends, in terms of y is y2 , and the total distance around the inside edge of the two semicircular parts of the track πy .

Explanation of Solution

Given:

The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end is shown in Figue-1 in part (a).

Concept used:

The length of the diameter and circumference of a circle of radius“r ”are given by 2r and 2πr respectively.

Calculations:

From the Figure-1, the diameter of each semi-circle is y meter, i.e.

2r=y

r=y2 , the radius of each semicircular end is y2 .

The distance in terms of y , around the inside edge of the each semicircular parts of the track is half circumference = πr=πy2 .

The total distance in terms of y , around the inside edge of the two semicircular parts of the track is half circumference = πy2+πy2=πy .

Conclusion:

The radius of semicircular ends, in terms of y is y2 , and the total distance around the inside edge of the two semicircular parts of the track πy .

To determine

To write:An equation in terms of x and y , for the distance travelled in one lap around the track, and solve it for y .

Expert Solution

Answer to Problem 63E

The equation for total length of the track is πy+2x=L , and its solution for yis y=2(100−x)π .

Explanation of Solution

Given:

The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end is shown in Figue-1 in part (a).

Formula used:

The formulae obtained in part (b).

Calculations:

From the Figure-1 in part (a), the total length L (say) of the track is

L=πr+x+πr+x=2πr+2x

⇒L=2πy2+2x=πy+2x

But given L=200 - meter.

⇒πy+2x=200

Thus, the equation for total length of the track is πy+2x=L , and its solution for y is y=2(100−x)π

Conclusion:

The equation for total length of the track is πy+2x=L , and its solution for y is y=2(100−x)π

To determine

To write: An expression for the area A of rectangular region of the room as a function of x .

Expert Solution

Answer to Problem 63E

The equation for total length of the track is πy+2x=L , and its solution for yis y=2(100−x)π .

Explanation of Solution

Given:

The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end is shown in Figue-1 in part (a).

Formula used:

The formulae obtained in part (b).

Calculations:

The areaA of the rectangular region of the physical fitness room is

A=x⋅y=2x(100−x)π

Thus, area A as a function of xis A(x)=2x(100−x)π .

Conclusion:

The area A as a function of x is A(x)=2x(100−x)π .

To determine

To graph: The area A of rectangular region of the room as a function of x , and approximate the dimension so that area A is maximum.

Expert Solution

Answer to Problem 63E

The graph of area A as a function of x is shown in Figure-2, and maximum area of the rectangular region can be A(x)max.=1591.55 square meter when dimensions of rectangle are x=50 meter and y=100π meter.

Explanation of Solution

Given:

The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end is shown in Figue-1 in part (a).

Formula used:

The formulae obtained in part (d).

Calculations:

The areaA of the rectangular region of the physical fitness room is

A(x)=2x(100−x)π

The graph of area A as a function of x is shown in figure here.

Precalculus with Limits: A Graphing Approach, Chapter 2.1, Problem 63E , additional homework tip 2

From the graph of A(x) we see that the area of the rectangular region is maximum when x=50 meter and y=2(100−x)π=2(100−50)π=100π meter.

Thus, the maximum area A(x)max.=1591.55 square meter.

Conclusion:

Chapter 2.1, Problem 62E

Chapter 2.1, Problem 64E

Chapter 2 Solutions

Precalculus with Limits: A Graphing Approach

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Prob. 28E Ch. 2.7 - Prob. 29E Ch. 2.7 - Prob. 30E Ch. 2.7 - Prob. 31E Ch. 2.7 - Prob. 32E Ch. 2.7 - Prob. 33E Ch. 2.7 - Prob. 34E Ch. 2.7 - Prob. 35E Ch. 2.7 - Prob. 36E Ch. 2.7 - Prob. 37E Ch. 2.7 - Prob. 38E Ch. 2.7 - Prob. 39E Ch. 2.7 - Prob. 40E Ch. 2.7 - Prob. 41E Ch. 2.7 - Prob. 42E Ch. 2.7 - Prob. 43E Ch. 2.7 - Prob. 44E Ch. 2.7 - Prob. 45E Ch. 2.7 - Prob. 46E Ch. 2.7 - Prob. 47E Ch. 2.7 - Prob. 48E Ch. 2.7 - Prob. 49E Ch. 2.7 - Prob. 50E Ch. 2.7 - Prob. 51E Ch. 2.7 - Prob. 52E Ch. 2.7 - Prob. 53E Ch. 2.7 - Prob. 54E Ch. 2.7 - Prob. 55E Ch. 2.7 - Prob. 56E Ch. 2.7 - Prob. 57E Ch. 2.7 - Prob. 58E Ch. 2.7 - Prob. 59E Ch. 2.7 - Prob. 60E Ch. 2.7 - Prob. 61E Ch. 2.7 - Prob. 62E Ch. 2.7 - Prob. 63E Ch. 2.7 - Prob. 64E Ch. 2.7 - Prob. 65E Ch. 2.7 - Prob. 66E Ch. 2.7 - Prob. 67E Ch. 2.7 - Prob. 68E Ch. 2.7 - Prob. 69E Ch. 2.7 - Prob. 70E Ch. 2.7 - Prob. 71E Ch. 2.7 - Prob. 72E Ch. 2.7 - Prob. 73E Ch. 2.7 - Prob. 74E Ch. 2.7 - Prob. 75E Ch. 2.7 - Prob. 76E Ch. 2.7 - Prob. 77E Ch. 2.7 - Prob. 78E Ch. 2.7 - Prob. 79E Ch. 2.7 - Prob. 80E Ch. 2.7 - Prob. 81E Ch. 2.7 - Prob. 82E Ch. 2.7 - Prob. 83E Ch. 2.7 - Prob. 84E Ch. 2.7 - Prob. 85E Ch. 2.7 - Prob. 86E Ch. 2.7 - Prob. 87E Ch. 2.7 - Prob. 88E Ch. 2.7 - Prob. 89E Ch. 2.7 - Prob. 90E Ch. 2.7 - Prob. 91E Ch. 2.7 - Prob. 92E Ch. 2.7 - Prob. 93E Ch. 2.7 - Prob. 94E Ch. 2.7 - Prob. 95E Ch. 2.7 - Prob. 96E Ch. 2.7 - Prob. 97E Ch. 2.7 - Prob. 98E Ch. 2.7 - Prob. 99E Ch. 2.7 - Prob. 100E Ch. 2.7 - Prob. 101E Ch. 2.7 - Prob. 102E Ch. 2.7 - Prob. 103E Ch. 2.7 - Prob. 104E Ch. 2.7 - Prob. 105E Ch. 2.7 - Prob. 106E Ch. 2.7 - Prob. 107E Ch. 2.8 - Prob. 1E Ch. 2.8 - Prob. 2E Ch. 2.8 - Prob. 3E Ch. 2.8 - Prob. 4E Ch. 2.8 - Prob. 5E Ch. 2.8 - Prob. 6E Ch. 2.8 - Prob. 7E Ch. 2.8 - Prob. 8E Ch. 2.8 - Prob. 9E Ch. 2.8 - Prob. 10E Ch. 2.8 - Prob. 11E Ch. 2.8 - Prob. 12E Ch. 2.8 - Prob. 13E Ch. 2.8 - Prob. 14E Ch. 2.8 - Prob. 15E Ch. 2.8 - Prob. 16E Ch. 2.8 - Prob. 17E Ch. 2.8 - Prob. 18E Ch. 2.8 - Prob. 19E Ch. 2.8 - Prob. 20E Ch. 2.8 - Prob. 21E Ch. 2.8 - Prob. 22E Ch. 2.8 - Prob. 23E Ch. 2.8 - Prob. 24E Ch. 2.8 - Prob. 25E Ch. 2.8 - Prob. 26E Ch. 2.8 - Prob. 27E Ch. 2.8 - Prob. 28E Ch. 2.8 - Prob. 29E Ch. 2.8 - Prob. 30E Ch. 2.8 - Prob. 31E Ch. 2.8 - Prob. 32E Ch. 2.8 - Prob. 33E Ch. 2.8 - Prob. 34E Ch. 2.8 - Prob. 35E Ch. 2.8 - Prob. 36E Ch. 2.8 - Prob. 37E Ch. 2.8 - Prob. 38E Ch. 2 - Prob. 1RE Ch. 2 - Prob. 2RE Ch. 2 - Prob. 3RE Ch. 2 - Prob. 4RE Ch. 2 - Prob. 5RE Ch. 2 - Prob. 6RE Ch. 2 - Prob. 7RE Ch. 2 - Prob. 8RE Ch. 2 - Prob. 9RE Ch. 2 - Prob. 10RE Ch. 2 - Prob. 11RE Ch. 2 - Prob. 12RE Ch. 2 - Prob. 13RE Ch. 2 - Prob. 14RE Ch. 2 - Prob. 15RE Ch. 2 - Prob. 16RE Ch. 2 - Prob. 17RE Ch. 2 - Prob. 18RE Ch. 2 - Prob. 19RE Ch. 2 - Prob. 20RE Ch. 2 - Prob. 21RE Ch. 2 - Prob. 22RE Ch. 2 - Prob. 23RE Ch. 2 - Prob. 24RE Ch. 2 - Prob. 25RE Ch. 2 - Prob. 26RE Ch. 2 - Prob. 27RE Ch. 2 - Prob. 28RE Ch. 2 - Prob. 29RE Ch. 2 - Prob. 30RE Ch. 2 - Prob. 31RE Ch. 2 - Prob. 32RE Ch. 2 - Prob. 33RE Ch. 2 - Prob. 34RE Ch. 2 - Prob. 35RE Ch. 2 - Prob. 36RE Ch. 2 - Prob. 37RE Ch. 2 - Prob. 38RE Ch. 2 - Prob. 39RE Ch. 2 - Prob. 40RE Ch. 2 - Prob. 41RE Ch. 2 - Prob. 42RE Ch. 2 - Prob. 43RE Ch. 2 - Prob. 44RE Ch. 2 - Prob. 45RE Ch. 2 - Prob. 46RE Ch. 2 - Prob. 47RE Ch. 2 - Prob. 48RE Ch. 2 - Prob. 49RE Ch. 2 - Prob. 50RE Ch. 2 - Prob. 51RE Ch. 2 - Prob. 52RE Ch. 2 - Prob. 53RE Ch. 2 - Prob. 54RE Ch. 2 - Prob. 55RE Ch. 2 - Prob. 56RE Ch. 2 - Prob. 57RE Ch. 2 - Prob. 58RE Ch. 2 - Prob. 59RE Ch. 2 - Prob. 60RE Ch. 2 - Prob. 61RE Ch. 2 - Prob. 62RE Ch. 2 - Prob. 63RE Ch. 2 - Prob. 64RE Ch. 2 - Prob. 65RE Ch. 2 - Prob. 66RE Ch. 2 - Prob. 67RE Ch. 2 - Prob. 68RE Ch. 2 - Prob. 69RE Ch. 2 - Prob. 70RE Ch. 2 - Prob. 71RE Ch. 2 - Prob. 72RE Ch. 2 - Prob. 73RE Ch. 2 - Prob. 74RE Ch. 2 - Prob. 75RE Ch. 2 - Prob. 76RE Ch. 2 - Prob. 77RE Ch. 2 - Prob. 78RE Ch. 2 - Prob. 79RE Ch. 2 - Prob. 80RE Ch. 2 - Prob. 81RE Ch. 2 - Prob. 82RE Ch. 2 - Prob. 83RE Ch. 2 - Prob. 84RE Ch. 2 - Prob. 85RE Ch. 2 - Prob. 86RE Ch. 2 - Prob. 87RE Ch. 2 - Prob. 88RE Ch. 2 - Prob. 89RE Ch. 2 - Prob. 90RE Ch. 2 - Prob. 91RE Ch. 2 - Prob. 92RE Ch. 2 - Prob. 93RE Ch. 2 - Prob. 94RE Ch. 2 - Prob. 95RE Ch. 2 - Prob. 96RE Ch. 2 - Prob. 97RE Ch. 2 - Prob. 98RE Ch. 2 - Prob. 99RE Ch. 2 - Prob. 100RE Ch. 2 - Prob. 101RE Ch. 2 - Prob. 102RE Ch. 2 - Prob. 103RE Ch. 2 - Prob. 104RE Ch. 2 - Prob. 105RE Ch. 2 - Prob. 106RE Ch. 2 - Prob. 107RE Ch. 2 - Prob. 108RE Ch. 2 - Prob. 109RE Ch. 2 - Prob. 110RE Ch. 2 - Prob. 111RE Ch. 2 - Prob. 112RE Ch. 2 - Prob. 113RE Ch. 2 - Prob. 114RE Ch. 2 - Prob. 115RE Ch. 2 - Prob. 116RE Ch. 2 - Prob. 117RE Ch. 2 - Prob. 118RE Ch. 2 - Prob. 119RE Ch. 2 - Prob. 120RE Ch. 2 - Prob. 121RE Ch. 2 - Prob. 122RE Ch. 2 - Prob. 123RE Ch. 2 - Prob. 124RE Ch. 2 - Prob. 125RE Ch. 2 - Prob. 126RE Ch. 2 - Prob. 127RE Ch. 2 - Prob. 128RE Ch. 2 - Prob. 129RE Ch. 2 - Prob. 130RE Ch. 2 - Prob. 131RE Ch. 2 - Prob. 132RE Ch. 2 - Prob. 133RE Ch. 2 - Prob. 134RE Ch. 2 - Prob. 135RE Ch. 2 - Prob. 136RE Ch. 2 - Prob. 137RE Ch. 2 - Prob. 138RE Ch. 2 - Prob. 139RE Ch. 2 - Prob. 140RE Ch. 2 - Prob. 141RE Ch. 2 - Prob. 142RE Ch. 2 - Prob. 143RE Ch. 2 - Prob. 144RE Ch. 2 - Prob. 145RE Ch. 2 - Prob. 146RE Ch. 2 - Prob. 147RE Ch. 2 - Prob. 148RE Ch. 2 - Prob. 149RE Ch. 2 - Prob. 150RE Ch. 2 - Prob. 151RE Ch. 2 - Prob. 152RE Ch. 2 - Prob. 153RE Ch. 2 - Prob. 154RE Ch. 2 - Prob. 155RE Ch. 2 - Prob. 156RE Ch. 2 - Prob. 157RE Ch. 2 - Prob. 158RE Ch. 2 - Prob. 159RE Ch. 2 - Prob. 160RE Ch. 2 - Prob. 161RE Ch. 2 - Prob. 162RE Ch. 2 - Prob. 163RE Ch. 2 - Prob. 164RE Ch. 2 - Prob. 165RE Ch. 2 - Prob. 166RE Ch. 2 - Prob. 167RE Ch. 2 - Prob. 168RE Ch. 2 - Prob. 1CT Ch. 2 - Prob. 2CT Ch. 2 - Prob. 3CT Ch. 2 - Prob. 4CT Ch. 2 - Prob. 5CT Ch. 2 - Prob. 6CT Ch. 2 - Prob. 7CT Ch. 2 - Prob. 8CT Ch. 2 - Prob. 9CT Ch. 2 - Prob. 10CT Ch. 2 - Prob. 11CT Ch. 2 - Prob. 12CT Ch. 2 - Prob. 13CT Ch. 2 - Prob. 14CT Ch. 2 - Prob. 15CT Ch. 2 - Prob. 16CT Ch. 2 - Prob. 17CT Ch. 2 - Prob. 18CT Ch. 2 - Prob. 19CT Ch. 2 - Prob. 20CT Ch. 2 - Prob. 21CT Ch. 2 - Prob. 22CT Ch. 2 - Prob. 23CT

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To draw: The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end.

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To draw:The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end.

Explanation of Solution

The radius of semicircular ends, and the distance in terms of y , around the inside edge of the two semicircular parts of the track.

Answer to Problem 63E

Explanation of Solution

To write:An equation in terms of x and y , for the distance travelled in one lap around the track, and solve it for y .

Answer to Problem 63E

Explanation of Solution

To write: An expression for the area A of rectangular region of the room as a function of x .

Answer to Problem 63E

Explanation of Solution

To graph: The area A of rectangular region of the room as a function of x , and approximate the dimension so that area A is maximum.

Answer to Problem 63E

Explanation of Solution

Chapter 2 Solutions