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 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 of $ A $ the rectangular region as a function of $ x $ What dimensions 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 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 of $ A $ the rectangular region as a function of $ x $ What dimensions will produce a rectangle of maximum area?

Key Concepts

Composite Figures

This concept involves breaking down a complex shape into simpler, standard geometric shapes such as rectangles, circles, or semicircles. When a figure is composed of several basic shapes, understanding how each contributes to the overall properties—like area and perimeter—allows for systematic analysis and calculation.

Perimeter Calculation

Perimeter calculation is the process of summing the lengths of all the outer edges of a shape. For composite figures, this requires calculating the contributions of both linear segments (such as those from a rectangle) and curved segments (from arcs or semicircular ends) using the appropriate formulas for each part.

Circle Geometry and Semicircles

This key concept focuses on the properties of circles and their parts, such as the circumference (which is calculated as 2?r for a full circle) and, by extension, arcs and semicircles. Understanding that a semicircle represents half of a full circle is crucial for determining its curved edge length and radius.

Algebraic Relationships in Geometry

Establishing relationships between different geometric dimensions (such as lengths and widths) using algebra is essential. By setting up equations based on constraints (like a fixed perimeter), one can express one variable in terms of another, creating a system that can be solved to find the dimensions of the shape.

Optimization in Geometry

Optimization in geometry involves finding the maximum or minimum values of a quantity—often area or volume—under a given set of constraints. This frequently requires formulating the quantity as a function of one variable and then applying techniques from calculus or algebra to determine the conditions for which the function attains its maximum or minimum.

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