Quadratic Functions) An indoor physical fitness room...

Quadratic Functions) 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.

Quadratic Functions) 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

Composite Shapes

Composite shapes are figures constructed from more than one basic geometric shape, such as rectangles and circles. Understanding composite shapes is crucial for determining properties like area and perimeter, as one needs to combine the standard formulas for each simple shape appropriately to analyze the overall figure.

Perimeter of Composite Figures

Calculating the perimeter of a composite figure involves summing the lengths of its individual components. For example, when a rectangle is complemented by semicircular ends, the perimeter is obtained by adding the lengths of the rectangle's sides and the arc lengths of the semicircles, while carefully accounting for overlaps or shared boundaries between the components.

Quadratic Functions in Optimization

Quadratic functions often emerge in optimization problems where a certain physical quantity, such as area, is to be maximized or minimized subject to some constraint. By formulating a relationship between the dimensions of a figure and a fixed perimeter or other total measure, one can derive a quadratic function whose maximum or minimum value yields the optimal dimensions.

Graphing and Analyzing Functions

Graphing functions is a powerful tool for visualizing relationships between variables and identifying key features such as vertices, intercepts, and extrema. In optimization problems, plotting the quadratic function representing the area helps to confirm the algebraic solution for the maximum area and allows for an approximate determination of the dimensions that achieve this optimum.

Transcript

00:01 Today, we are going to solve a problem related to the area and length of a rectangular track with semicircling either end.

00:06 It is five parts to it.

00:08 We will need to draw a diagram that illustrates the problem, figure out the radius of the semicircles on either end, use that to determine the area of the rectangular region, can write the area of the distance traveled in one lap around the track with that equation, and then graph the whole thing with a graphing calculator to find the optimal area.

00:22 Part b, we can skip because that's just drawing a graph, and i'll assume you know how to do that by now.

00:28 Part b is where things start getting complicated.

00:30 The straight rectangular portion represents the length.

00:34 I think so if y is the width, if the meters, the distance around the same circle can be found by multiplying, buying it by pi.

00:39 Because the distance around a circle is 2 pi times r, where r is the radius.

00:43 In this case, y is the radius.

00:45 However, because there's only half a circle, we only need half of that.

00:49 So we get pi y.

00:51 So for the next part, we will need to relate the entire equation to pi y.

00:58 So we need to figure out, okay, how did the straight -oids work? it's actually rather simple.

01:03 Because the maximum length of the track is 200 meters, it means a single straightaway can't be more than 100 meters...

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