Concept explainers
a.
To draw:The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end.
a.
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
Drawing:
The diagram of indoor physical fitness rom consisting a rectangular region with semicircle on each end is shown in Figue-1 here.
b.
The radius of semicircular ends, and the distance in terms of y , around the inside edge of the two semicircular parts of the track.
b.
Answer to Problem 63E
The radius of semicircular ends, in terms of y is
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
Calculations:
From the Figure-1, the diameter of each semi-circle is y meter, i.e.
The distance in terms of y , around the inside edge of the each semicircular parts of the track is half circumference =
The total distance in terms of y , around the inside edge of the two semicircular parts of the track is half circumference =
Conclusion:
The radius of semicircular ends, in terms of y is
c.
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 .
c.
Answer to Problem 63E
The equation for total length of the track is
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
But given
Thus, the equation for total length of the track is
Conclusion:
The equation for total length of the track is
d.
To write: An expression for the area A of rectangular region of the room as a function of x .
d.
Answer to Problem 63E
The equation for total length of the track is
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
Thus, area A as a function of xis
Conclusion:
The area A as a function of x is
e.
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.
e.
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
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
The graph of area A as a function of x is shown in figure here.
From the graph of
Thus, the maximum area
Conclusion:
The graph of area A as a function of x is shown in Figure-2, and maximum area of the rectangular region can be
Chapter 2 Solutions
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