Geometry An indoor physical fitness room consists of a...

Geometry An indoor physical fitness room consists of a rectangular region with a semicircle on each end.The perimeter of the room is to be a 200 -meter single-lane running track. (a) Draw a diagram that gives a visual representation of the problem. Let $x$ and $y$ represent the length and width of the rectangular region, respectively. (b) Determine the radius of each semicircular end of the room. Determine the distance, in terms of $y,$ around the inside edge of each semicircular part of the track. (c) Use the result of part (b) to write an equation, in terms of $x$ and $y,$ for the distance traveled in one lap around the track. Solve for $y .$ (d) Use the result of part (c) to write the area $A$ of the rectangular region as a function of $x .$ What dimensions will produce a rectangle of maximum area?

Geometry An indoor physical fitness room consists
of a rectangular region with a semicircle on each end.The perimeter of the room is to be a 200 -meter
single-lane running track.
(a) Draw a diagram that gives a visual representation of
the problem. Let $x$ and $y$ represent the length and
width of the rectangular region, respectively.
(b) Determine the radius of each semicircular end of the
room. Determine the distance, in terms of $y,$ around
the inside edge of each semicircular part of the track.
(c) Use the result of part (b) to write an equation, in
terms of $x$ and $y,$ for the distance traveled in one lap
around the track. Solve for $y .$
(d) Use the result of part (c) to write the area $A$ of
the rectangular region as a function of $x .$ What
dimensions will produce a rectangle of maximum
area?

Key Concepts

Composite Figures

This concept involves analyzing shapes that are made up of more than one simple geometric figure. In this problem, the fitness room is represented by a rectangle with semicircular ends. Understanding composite figures is essential to correctly derive relationships for perimeter and area by considering each component separately and then combining the results.

Perimeter Calculation

Calculating the perimeter of composite figures requires summing the lengths of straight sides and curved portions. In this scenario, the total track length is made up of the rectangle’s sides and the arcs of the semicircles. This concept is crucial for setting up equations that relate the dimensions of the individual parts to the overall perimeter.

Variable Representation and Equation Setup

Using variables to denote dimensions (like x for length and y for width) allows the establishment of algebraic relationships. This approach facilitates forming equations based on geometric properties, such as the relation between the width of the rectangle and the radius of the semicircular ends, ultimately linking the perimeter and area expressions.

Relationship Between Variables

In geometric problems, the dimensions of composite parts are often interdependent. For instance, the width of the rectangle determines the diameter (and therefore the radius) of the semicircular sections. Recognizing these relationships is key to expressing one variable in terms of others, simplifying the problem to a single-variable analysis for optimization.

Function Optimization

Optimization involves determining the best (maximum or minimum) values of a function under given constraints. In this problem, the area of the rectangle is optimized by expressing it as a function of one variable and then using techniques such as calculus or algebraic methods to find the maximum area. This is an essential step in many geometric and applied mathematics problems.

Diagram Visualization

Visual representation through diagrams helps in understanding the spatial arrangement of the composite figure. Drawing a diagram to illustrate the rectangle with attached semicircles aids in accurately defining the variables, visualizing the problem, and formulating the correct perimeter and area equations.

Transcript

00:01 We're asked to answer a question about geometry using quadratic functions and models.

00:05 What are you looking at, adam? two guys on the phone at once? absolutely not.

00:10 I'm not looking at an accurate comment.

00:12 He has a very unreadies.

00:14 There's no freddie mercury, but still, rod stewart is a great place.

00:18 In part a, okay, we're given that an indoor physical fitness room consists of a rectangular region with a semicircle on each end.

00:26 The perimeter of the room is a 200 meter, around the room is a 200 meter single lane running track.

00:35 And in part a, we're asked to draw a diagram that gives a visual representation of this problem.

00:40 And we're supposed to let x and y represent the length and width of the rectangular region, respectively.

00:48 Well, so we have a rectangular region, which has a length of x and a width of y.

00:58 And then around this rectangular region, where we have semicircles on both ends, like this.

01:11 And the perimeter of the whole thing, p, this is 200 meters.

01:22 We'll just assume this running track has a thickness of zero for simplicity.

01:42 Then in part b, we're asking the radius of each semicircular end of the room.

01:49 We're asking to determine the radius of each semicircular end of the room.

02:03 We're also asking to determine the distance in terms of why around the inside edge of each semicircular part of the track.

02:19 Okay, so the radius of the semicircle, which we can call r, is very clearly half of diameter, which is y...

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