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

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 illustrates the problem. Let $x$ and $y$ represent the length and width of the rectangular region, respectively. (b) Determine the radius of the semicircular ends of the track. Determine the distance, in terms of $y$, around the inside edge of the two semicircular parts 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 $x$ (e) Use a graphing utility to graph the area function from part (d). Use the graph to approximate the dimensions that will produce a rectangle of maximum area.

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 illustrates the problem. Let $x$ and $y$ represent the length and width of the rectangular region, respectively.
(b) Determine the radius of the semicircular ends of the track. Determine the distance, in terms of $y$, around the inside edge of the two semicircular parts 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 $x$
(e) Use a graphing utility to graph the area function from part (d). Use the graph to approximate the dimensions that will produce a rectangle of maximum area.

Key Concepts

Function Modeling

Function modeling is the practice of expressing one physical quantity, such as the area, as a function of one or more variables related to the dimensions of the shape. This allows for the analysis and optimization of the quantity under specified constraints.

Constraint Equations

This concept refers to setting up mathematical relationships between different dimensions of a geometric figure using given conditions. In problems like this, the known total perimeter is used to express one variable in terms of another, thereby reducing the number of independent variables in the problem.

Graphical Optimization

Graphical optimization involves using graphing techniques or numerical tools to visualize the behavior of a function and identify its maximum or minimum points. In this context, it is used to determine the dimensions that maximize the area of the rectangle, based on the relationship derived from the geometric constraints.

Composite Geometry

This concept involves understanding how to represent and analyze shapes that are made up of simpler geometric figures. In this problem, a rectangle with semicircular ends is used, which requires combining properties of both rectangles and circles in order to determine measurements like area and perimeter.

Perimeter Calculation

This key concept focuses on calculating the total boundary length of a composite figure. It involves summing the lengths of straight segments (from the rectangle) and curved segments (from the semicircles) to form an overall perimeter constraint for the design.

Transcript

00:03 So, in this question it is given to me that there is a rectangular area around which semicircles are drawn.

00:11 So i have this rectangular area whose length, let us call this one as length x and this one let it be y.

00:20 So mark it, this is y and this length which is coming out, this is x units.

00:29 So that is how the figure is.

00:32 And in the next part they are asking me determine the radius of the semicircular ends.

00:36 So radius of semicircular ends are, i can observe here that the length x and the length y are the diameter of the semicircle.

01:05 So radius of the semicircle will be half of the diameter.

01:08 So that will be x by 2 and y by 2.

01:13 All right.

01:14 Now they are asking me determine the distance in terms of y around the inside edge of the two semicircular parts of the track.

01:20 That means this one they are asking, right? this inside edge of the semicircle.

01:27 So total distance in terms of y around the inside edge of around inside edge of semicircle with radius equal to y by 2.

01:56 So the two part, the two semicircles whose radius are y by 2 that will be equal to 2 pi r.

02:04 Now what is r over here? that is y by 2.

02:07 So that total distance will be pi y.

02:10 All right.

02:11 Similarly, so this is the b part...

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